В раздевалке я положил сумку в шкафчик и быстро переоделся.

Because it expresses location: in the changing room / locker room.

After в and на, Russian often uses:

• prepositional for where?
• accusative for where to?

So:

• в раздевалке = in the locker room → location
• в шкафчик = into the locker → direction/destination

The noun changes like this:

• раздевалка → в раздевалке

This is a very common pattern in Russian.

Because сумку is the accusative singular form of сумка.

The verb положить means to put/place, and it takes a direct object: you put something. That something usually goes in the accusative case.

So:

• сумка = bag
• положил сумку = put the bag

This is the same idea as in English I put the bag...

Because here в means into, not in.

Russian uses:

• в + accusative for movement into
• в + prepositional for being inside

So:

• в шкафчик = into the locker
• в шкафчике = in the locker

Since the bag is being moved into the locker, Russian uses the accusative: шкафчик.

Шкафчик is a diminutive form of шкаф.

• шкаф = wardrobe, cabinet, closet
• шкафчик = little cabinet, small locker, locker

In this context, шкафчик is the natural word for a gym/school/changing-room locker.

The -чик ending often makes a noun sound smaller or more specific.

Положил is the perfective past form of положить, which focuses on a completed action.

Here the speaker is describing a sequence of finished actions:

1. put the bag in the locker
2. changed clothes

So perfective fits very well:

• положил = put / placed, completed
• клал or положил?
  
  • клал would be imperfective and is less natural here if you simply mean the action was completed once
  • положил is the normal choice for a one-time completed event

Russian often uses perfective verbs in narration like this.

Because переодеться means to change one’s clothes / get changed, and it is normally used as a reflexive verb.

• переодеть кого-то = to change someone else’s clothes
• переодеться = to change one’s own clothes

So:

• я быстро переоделся = I quickly changed clothes / got changed quickly

The -ся shows that the action is directed back to the subject.

That is just how many reflexive verbs look in the past tense.

The infinitive is:

• переодеться

The masculine past form is:

• переоделся

This breaks down as:

• переодел = changed
• ся = reflexive marker

When added together, you get переоделся.

This pattern is very common:

• умыться → умылся
• одеться → оделся
• вернуться → вернулся

In the past tense, Russian verbs agree with the subject in gender and number.

Here:

• положил
• переоделся

show that the speaker is male.

If the speaker were female, it would be:

• В раздевалке я положила сумку в шкафчик и быстро переоделась.

So past-tense verbs in Russian tell you something about the speaker that English verbs usually do not.

It can mean either, depending on context.

Common translations include:

• changing room
• locker room
• cloakroom in some contexts, though that is less likely here

Because the sentence mentions a locker and changing clothes, locker room or changing room is the best fit.

That is the normal position for an adverb modifying the verb.

• быстро переоделся = quickly changed clothes

Russian word order is flexible, but this placement is very natural and neutral.

You could move words around for emphasis, for example:

• я переоделся быстро
• быстро я переоделся

But the original version sounds the most straightforward in ordinary narration.

Yes, Russian can often omit subject pronouns when the verb already makes the subject clear.

So this would also be natural:

• В раздевалке положил сумку в шкафчик и быстро переоделся.

However, including я is also completely normal. It may help with:

• clarity
• contrast
• a slightly more explicit narrative style

So both are possible; the version with я is not strange.

Yes. That is one of the key things the sentence shows.

Both main verbs are perfective past:

• положил
• переоделся

That gives the sense of finished, single actions in sequence:

• I put the bag in the locker,
• and then I quickly changed clothes.

This is a very typical storytelling pattern in Russian.