How to become more articulate?

I spent four years at toastmasters and wouldn't really recommend it if your goal is to improve your normal conversation skills as opposed to stage performance. 

The problem with Toastmasters is that you spent most of your time at a toastmasters meeting listening to other people who talk.

Improv Comedy is an alternative that gives you much more practice time at talking. It's also more targeted at the conversation skill of coming up with things to say in a conversation. 

1. Write more. Writing will help you formulate your thinking which helps in speaking. You'll also find yourself building your vocabulary.
2. Go through an online public speaking course.
3. Join a Toastmasters club.
4. Make Youtube videos
5. Watch podcast interviews. Pay attention to how the host asks questions. 
   
   Instead of saying "Please talk about X for the next 2 minutes so I can absorb your knowledge." try "What do you think are the most important concepts in your field? or "What do you think are the biggest misconceptions about your field?" etc.
6. Take a step back and identify your terminal goals. 
   
   Are you trying to build more self-confidence? 
   
   Are you trying to get over social anxiety? There might be more efficient/upstream ways to solve this like experimenting with supplements, beta-blockers, phenibut, etc. 
   
   Are you trying to be more persuasive? Maybe persuasion through conversation isn't your comparative advantage and you should focus on something else?
   
   
    

Study, don't just watch, good debates or speeches.  Recognize how much prep went into them.  Listen carefully to podcasts you admire, and again think about how they and their guests prepare.  

Slow down.  Take 2-5 seconds to plan before starting a sentence.  You may need filler words here, just to signal that you're thinking and not passing your turn.

Intentional practice.  Notice when you're less precise than you'd hoped, and fix it.  Many (not all) conversations with friends or coworkers have room for "wait, let me restate that" and try again.  Do this in all situations where you think this kind of precision is a primary component of the communication.  And notice which conversations would not benefit, so you can be more natural and off-the-cuff in those.

Read a lot.  Vocabulary comes from outside - you'll get used to the words you encounter often.  Writing is also critical - you can do the thinking and editing before anyone sees it, and this will make you more articulate when speaking.  Also, many people who are conversationally articulate heavily use cached thoughts that they've refined in writing.

For asking questions, I think you've got it backward.  You may wish to have someone give you a 2-minute primer on some topic you're interested in, but that's rarely easy or fun for them.  Be up-front about your level of knowledge and areas of curiosity, but try to ask specific questions that fill in gaps, rather than "please work hard to give me a jumpstart on a topic you've thought about for years".

I want to double down on write more. Writing more has helped be become a better speaker, mostly for the reasons mentioned elsewhere in this thread, such as learning to formulate complete thoughts and structure thinking so that one sentence connects to another to form a chain of reasoning.

One thing I haven't seen mentioned is learning to shut up. Shut up. Even when it means that there will be an awkward silence. Let your thoughts form fully, and only after they're complete, speak. Remember the ancient proverb: 'tis better to keep silent and be thought a fool than to open one's mouth and remove all doubt. In my experience, a significant portion of people seeming inarticulate comes from their rushing to get their thoughts out of their head in order to fill "dead air".

Regarding eliminating filler-words, My friend and I have a very effective strategy that we employ about once a year. In fact, we're just about to commence another round -- we called it "No-'um'-November". 

The rules are simple:

• Make a shortlist of words you want to eliminate. For me, they are typically "Um", "Like", "You know", and "Kind of". Don't pick too many.
• Every time you use the word, hit yourself hard on one cheek. A real firm, hopefully painful smack. It doesn't matter where you are, nor who you're talking to. No exceptions. Strike.

It is more enjoyable, and less odd, when you take this challenge with one or more friends or colleagues who you see on a daily basis.

The first time I tried this, I cut down nearly all filler-words within the first four days, and the rest of the month I spent simply being on high-alert.

Fortunately for us, we both have cheerful colleagues who are forgiving of "unusual" behaviour in the office.

Write in order to organize your thoughts, maybe read a business writing course for succinctness, then record yourself giving a short explanation of what you've learned about the topic. Go slowly as you need to to avoid filler. Watch the recording and process the emotions/discomforts with your speaking that come up via gendlin's focusing or core transformation.

Read or listen to a lot of the kind of material you hope to be able to produce yourself (people speaking clearly and eloquently), or really just a lot of things in general - anything longform whether it's essays or novels or non-fiction will help with vocabulary and style. 

Feed it all into the maw of a woodchipper inside your head, to be chewed up and absorbed into your own habits of speech and thought. The more contact you have with lots of good examples, the more you can draw on the patterns and rhythms of it, to generate more like it.

But also practice. Lots of occasions of speaking, either in front of people or in quiet moments to yourself. Make a conscious effort to be clear, until it becomes more of an unconscious skill. At first that will mean thinking carefully to plan what you intend to say, so that you can deliver it with fluency and fluidity (honestly, not just "at first" - very probably that will always help). With time maybe you get sufficiently accustomed to make the thinking phase shorter and speak more 'off the cuff', but being able to speak unprepared in flawless prose is a genuinely rare skill and perhaps not a realistic target.

There are also classes and clubs you could try, either specifically for public speaking and speechmaking, or for performance more generally (like theatre/improv).

You can use less weasel words.

Often I feel like the question I want to ask is

becomes

I want to ask:

and less passive voice

Being articulate" is a skill I feel like I do not have, and I would like to practice this skill."

becomes

I'm not articulate, and I want to practice being more articulate

You can stop using filler words by not using them! Know what you're going to say when you talk. If you don't know what to say next, don't say "um." You should stand there, quietly, until you have something to say. That is powerful. Listen to someone who has a refined speaking style like Ted Cruz and notice how he never says um. You should also watch the Cambridge Union debates and listen to how the more eloquent speakers speak in rehearsed phrases. The best way to sound smart is to spend hours preparing something and present it as if you made it up on the spot. Really smart people will have a ton of prepared phrases, so many that they can talk on a wide variety of topics by saying something they already know how to say and just modifying it a little.

Take time before you speak. If you're being asked about something, don't be afraid to say, "Let me think about that," or "That's a good question." These are essentially filler sentences of their own, but they are well articulated and they buy you time to think out what you're going to say next -- if you plan out what you're going to say, you won't need any filler words.

As for asking someone to transfer knowledge, phrasing is usually the key. Saying something like, "I'm trying to learn more about X. Can you give me an overview?" or, "I know you're more familiar with X than I am. Could you explain to me a little about how it works?"