После пробежки мне хочется выпить чего-нибудь посвежее, чем горячий чай.

Because после always requires the genitive case in Russian.

• пробежка = nominative singular
• пробежки = genitive singular

So:

• после пробежки = after a run

This is a fixed grammar pattern:

• после работы = after work
• после урока = after the lesson
• после тренировки = after training

Мне хочется is an impersonal way to say I feel like... or I want... in a more spontaneous, less direct sense.

Literally, it works like:

• мне = to me
• хочется = it is wanted / it feels desirable

So мне хочется выпить... means something like:

• I feel like drinking...
• I’d like to drink...

By contrast:

• я хочу = I want
• мне хочется = I feel like / I have a desire to

The second one often sounds softer and more natural when talking about a passing desire.

Because the construction хочется is impersonal and takes the experiencer in the dative.

In other words, Russian expresses this idea as something like:

• To me, it feels desirable...

That is why you get:

• мне хочется = I feel like
• ему хочется = he feels like
• нам хочется = we feel like

This is a very common pattern with words describing feelings or states:

• мне холодно = I am cold
• ей скучно = she is bored
• ему не спится = he can’t sleep

Because выпить is the perfective form, and here it suggests having a drink as a single, complete action.

• пить = to drink, to be drinking, to drink in general
• выпить = to drink up / to have a drink / to drink once

In this sentence, the speaker wants one completed action after the run:

• мне хочется выпить... = I feel like having a drink...

If you used пить, it would sound more like the process or general activity of drinking.

This is a very common question. After verbs of eating and drinking, Russian often uses the genitive to mean some amount of something.

So:

• что-нибудь = something
• чего-нибудь = some of something

With выпить, чего-нибудь sounds very natural because the speaker means something to drink, not just some random thing.

Compare:

• Я хочу что-нибудь купить. = I want to buy something.
• Я хочу выпить чего-нибудь. = I want to drink something.

This is related to the partitive-like use of the genitive, especially with food and drink.

Here it means something, specifically something to drink.

Even though the word itself is general, the verb выпить makes the meaning clear. So the listener understands:

• чего-нибудь = something drinkable

It is an indefinite pronoun:

• что-нибудь / чего-нибудь = something
• кто-нибудь = someone
• где-нибудь = somewhere

In this sentence, it has a slightly casual sense:

• something or other
• something to drink

Both are related to the comparative of свежий = fresh.

• свежее = fresher
• посвежее = a bit fresher / somewhat fresher / something on the fresher side

The prefix по- often softens the comparative and makes it sound less absolute, more conversational.

So:

• что-нибудь свежее = something fresher
• что-нибудь посвежее = something a bit fresher / something a little more refreshing

In this sentence, посвежее sounds very natural because the speaker is casually expressing a preference.

In this context, it most naturally means more refreshing.

With drinks, свежий can literally mean fresh, but in real usage it often overlaps with:

• cool
• refreshing
• not heavy or hot

So after a run, чего-нибудь посвежее, чем горячий чай suggests:

• something more refreshing than hot tea
• probably something cooler, lighter, or more thirst-quenching

So the idea is not only literal freshness, but also how the drink feels.

After a comparative, чем introduces the thing being compared, and that noun is often in the nominative case.

So:

• посвежее, чем горячий чай = fresher than hot tea

This is the standard comparison pattern:

• лучше, чем кофе = better than coffee
• интереснее, чем фильм = more interesting than the film

Russian also has another comparison pattern without чем, where the second item can appear in the genitive:

• умнее брата = smarter than his brother

But in your sentence, the structure with чем is the normal and clear choice.

Because it is the noun phrase following чем in a standard comparison.

So:

• горячий чай stays in its dictionary-like form
• adjective and noun agree in nominative singular masculine

That gives:

• горячий чай = hot tea

This happens because the phrase after чем is functioning as the comparison standard:

• more refreshing than hot tea

Russian word order is flexible, but this order is very natural.

• После пробежки sets the time/context first: After a run
• мне хочется gives the speaker’s feeling: I feel like
• выпить чего-нибудь... gives the desired action and object

So the sentence flows from:

1. situation
2. feeling
3. action
4. comparison

You could rearrange parts, but this version sounds smooth and idiomatic.

Most directly, it means after a run.

• пробежка is a noun meaning a run, a jog
• после пробежки = after the run / after a run

In English, depending on context, you might translate it more naturally as:

• After a run
• After jogging
• After going for a run

But grammatically in Russian, it is definitely a noun phrase, not a verb form.

Yes, it could, but the tone would change.

• Я хочу выпить чего-нибудь... = I want to drink something...
• Мне хочется выпить чего-нибудь... = I feel like drinking something...

The version with мне хочется sounds:

• softer
• more about a current craving
• more conversational

The version with я хочу sounds:

• more direct
• more straightforward
• sometimes a little stronger

So both are correct, but мне хочется fits the relaxed, after-exercise feeling very well.