Learn Turkish: Lütfen planı anlat. - Please explain the plan.

The suffix -ı marks it as a specific direct object, meaning "the plan" rather than just "a plan." Because you are telling someone to explain a specific thing, the noun takes this ending.

How does this sentence connect to the question word neyi (what specific)?

If you wanted to ask "What specific thing are you explaining?", you would use the question word neyi. The answer here is planı (the plan). Notice how both neyi and planı end in the same specific direct object vowel suffix!

How do I know whether to use -ı, -i, -u, or -ü for the specific object suffix here?

You use the 4-way vowel harmony rule. Since the last (and only) vowel in the root word plan is a, the correct matching suffix is -ı.

Why does the verb anlat not have any tense endings?

This is a direct command (the imperative form) addressed to one person informally. To tell someone to do something in Turkish, you simply use the verb root. You take the dictionary word anlatmak (to explain) and drop the -mak ending.