После прогулки в парке я чувствую себя бодрее.

Because после requires the genitive case.

• Dictionary form: прогулка
• Genitive singular: прогулки

So:

• после прогулки = after a walk
• после урока = after the lesson
• после работы = after work

This is a very common pattern in Russian: после + genitive.

Because here в парке answers the question where?, not to where?

Russian uses:

• в + accusative for motion into/to a place
  
  • идти в парк = to go to the park
• в + prepositional for location in/at a place
  
  • гулять в парке = to walk in the park

In this sentence, the walk happened in the park, so Russian uses the prepositional case:

• парк → в парке

Yes, я can often be omitted in Russian if the subject is clear from the verb.

• я чувствую = I feel
• чувствую by itself already shows I because of the verb ending -ю

So both are possible:

• После прогулки в парке я чувствую себя бодрее.
• После прогулки в парке чувствую себя бодрее.

Including я can make the sentence a bit clearer, more explicit, or slightly more natural depending on context.

In this expression, чувствовать себя means to feel / to feel oneself / to feel in a certain state.

So:

• чувствовать = to feel
• чувствовать себя = to feel (oneself), to feel a certain way

Examples:

• Я чувствую себя хорошо. = I feel good.
• Она чувствует себя плохо. = She feels bad.
• Мы чувствуем себя лучше. = We feel better.

English usually does not say I feel myself energetic, but Russian normally uses себя in this structure.

Себя is the accusative/genitive form of the reflexive pronoun себя.

With чувствовать себя, Russian uses the reflexive pronoun in a fixed pattern:

• я чувствую себя
• ты чувствуешь себя
• он чувствует себя
• мы чувствуем себя

Unlike English, Russian does not change this to something like myself / yourself / himself in the same way. The single reflexive pronoun себя is used for all persons.

Because after чувствовать себя, Russian often uses an adjective in the comparative to describe how someone feels.

• бодрый = energetic, vigorous
• бодрее = more energetic

So:

• чувствую себя бодрым would mean something slightly different grammatically, using the instrumental after a different pattern
• чувствую себя бодрее means I feel more energetic

In this sentence, the speaker is comparing their current state to an earlier or usual state.

Бодрее is the simple comparative form of бодрый.

Many Russian adjectives form the comparative with endings like:

• -ее / -ей
• sometimes with stem changes

Examples:

• быстрый → быстрее = faster
• умный → умнее = smarter
• бодрый → бодрее = more energetic

Russian often prefers this short comparative form where English uses more + adjective.

It can cover several related ideas, depending on context.

Бодрый often suggests:

• energetic
• refreshed
• lively
• alert
• less tired

So in this sentence, бодрее could mean something like:

• more energetic
• more refreshed
• more awake

It is not exactly the same as happier, though it can sometimes sound generally positive and lively.

Russian comparatives often leave the second part of the comparison unstated if it is obvious from context.

So я чувствую себя бодрее naturally means:

• I feel more energetic (than before)
• I feel more energetic now
• I feel more energetic than I usually do before the walk

If you want, Russian can make the comparison explicit:

• Я чувствую себя бодрее, чем раньше. = I feel more energetic than before.
• ...чем до прогулки. = ...than before the walk.

But it is very normal not to say this part.

Because the speaker is describing their current state now, after the walk.

• чувствую = I feel / I am feeling

So the timeline is:

1. the walk happened
2. now the speaker feels more energetic

Russian present tense works naturally here, just like English in:

• After a walk in the park, I feel better.

It can also sound like a general truth or habit, depending on context:

• After a walk in the park, I feel more energetic.

Yes. Russian word order is flexible, though different orders can sound slightly different in emphasis.

Original:

• После прогулки в парке я чувствую себя бодрее.

Possible alternatives:

• Я чувствую себя бодрее после прогулки в парке.
• Бодрее я чувствую себя после прогулки в парке. (more emphasis on бодрее)

The original order is very natural because it sets the scene first: after a walk in the park...

Mostly it is understood with прогулки:

• прогулки в парке = a walk in the park

So после прогулки в парке means after a walk in the park, not after a walk, while being in the park.

Russian often places location phrases right after the noun they describe, just as English does.

Yes, but it means something different.

• после прогулки в парке = after a walk in the park
  one specific walk, or a typical single walk
• после прогулок в парке = after walks in the park
  walks in general, or repeated walks

So the singular in the original sentence sounds natural if the speaker means one walk or the usual effect of taking a walk.

Yes, but the meaning becomes less specific.

• Я чувствую себя лучше. = I feel better.
• Я чувствую себя бодрее. = I feel more energetic / more refreshed.

So лучше is broader and more general, while бодрее tells you exactly how the person feels better.

Russian has no articles.

So в парке can mean:

• in the park
• in a park

The exact meaning depends on context. In many everyday sentences, context makes it clear whether a specific park is meant or not.

That is normal in Russian, and learners have to get used to understanding definiteness without a / an / the.