J’ai réussi à retrouver mes clés dans mon sac.

Because je drops its final e before a vowel sound. This is called elision.

• je + ai becomes j’ai
• French does this to make pronunciation smoother.

So:

• je ai réussi ❌
• j’ai réussi ✅

The same thing happens in many other phrases:

• j’aime
• j’habite
• j’arrive

It is the passé composé, a very common French past tense.

Here it is built with:

• ai = present tense of avoir
• réussi = past participle of réussir

So:

• j’ai réussi = I succeeded / I managed

In everyday French, the passé composé is often used where English might say:

• I managed to...
• I was able to...
• I succeeded in...

Because the verb réussir is commonly followed by à + infinitive when it means to manage to do something.

So the pattern is:

• réussir à faire quelque chose = to manage to do something

Examples:

• J’ai réussi à finir. = I managed to finish.
• Elle a réussi à partir tôt. = She managed to leave early.

So in your sentence:

• j’ai réussi à retrouver mes clés
  = I managed to find my keys again / I managed to locate my keys

Here, réussir à means to manage to or to succeed in doing something.

It does not usually translate literally as English to succeed to, because English does not say that.

So:

• J’ai réussi à retrouver mes clés
  = I managed to find my keys

This expression often suggests that the action was not easy, or that there was some difficulty before success.

This is a very common question.

• trouver = to find
• retrouver = to find again, recover, get back, locate after losing track of something

In this sentence, retrouver is very natural because keys are something you may have misplaced and then found again.

So:

• J’ai trouvé mes clés = I found my keys
• J’ai retrouvé mes clés = I found my keys again / I found my lost keys

In many everyday cases, English just says found, but French often prefers retrouver when the thing was lost or misplaced.

Yes, you could, and it would be understood.

But retrouvé sounds more natural if the keys were missing and you were looking for them.

Compare:

• J’ai trouvé mes clés dans mon sac
  = I found my keys in my bag
• J’ai retrouvé mes clés dans mon sac
  = I found my keys in my bag after having lost/misplaced them

So retrouver adds the idea of recovering something you had lost track of.

Because with avoir in the passé composé, the past participle usually does not agree with the subject.

So even if the subject is je, the form stays:

• j’ai réussi
• tu as réussi
• elle a réussi
• nous avons réussi

Agreement with a past participle used with avoir only happens in certain cases, mainly when a direct object comes before the participle. That is not happening here.

So réussi stays the same.

Because the speaker is talking about their own keys, so French uses a possessive adjective:

• mes = my
• clés = keys

So:

• mes clés = my keys

You could say les clés in some contexts, but that would just mean the keys, not specifically my keys.

Also note:

• mes is used with plural nouns
• singular would be:
  • ma clé = my key

Because sac is a masculine singular noun in French.

So the correct possessive adjective is:

• mon sac = my bag

Compare:

• mon sac = my bag
• ma clé = my key
• mes clés = my keys

French possessive adjectives agree with the thing possessed, not with the owner.

So it is:

• mon sac because sac is masculine singular
• mes clés because clés is plural

It tells you where the keys were found: in my bag.

So the idea is:

• I managed to find my keys
• where? → in my bag

In natural English, the phrase may seem to describe the whole action, but in French it works very naturally at the end of the sentence.

You can think of it as modifying the finding/recovering of the keys:

• retrouver mes clés dans mon sac = find my keys in my bag

Yes, sometimes, but the original order is very natural.

Standard sentence:

• J’ai réussi à retrouver mes clés dans mon sac.

You might also hear:

• J’ai réussi à retrouver mes clés, dans mon sac.
  This would usually sound more marked or spoken, with a pause.
• Dans mon sac, j’ai réussi à retrouver mes clés.
  This is possible, but it gives special emphasis to dans mon sac.

For a learner, the original version is the best model:

• subject + avoir + past participle + à + infinitive + object + place

Because after réussir à, French uses an infinitive.

So the structure is:

• j’ai réussi à + infinitive

That is why you get:

• j’ai réussi à retrouver
• not j’ai réussi à retrouve
• not j’ai réussi à retrouvé

The only conjugated verb in the second part is réussir, which appears here in the passé composé as j’ai réussi.
The next verb stays in the infinitive: retrouver.

Clés is pronounced roughly like clay in English, but with a pure French vowel.

The accent in clé is an accent aigu: é.

• clé = key
• clés = keys

You may also see clef / clefs, which is an older spelling still used sometimes.
Today, clé / clés is very common and standard.

The final s in clés is normally not pronounced.

A few small things:

• J’ai sounds like zhay
• réussi sounds roughly like ray-u-see
• à retrouver has a clear a sound in à
• mes clés sounds roughly like may clay
• dans mon sac has a nasal sound in dans

A rough pronunciation guide:

• J’ai réussi à retrouver mes clés dans mon sac
  ≈ zhay ray-u-see ah ruh-troo-vay may clay dahn mohn sak

Not perfect English-style phonetics, but enough to help you hear the parts.

The sentence follows a very useful pattern:

• Subject + avoir + past participle + à + infinitive + object + place

Here:

• J’ = subject
• ai réussi = passé composé of réussir
• à retrouver = infinitive phrase
• mes clés = object
• dans mon sac = place

So this sentence is a great model for building similar ones:

• J’ai réussi à finir mon travail.
• Elle a réussi à ouvrir la porte.
• Nous avons réussi à retrouver le document.

It is a very common and useful structure in French.