תודה שהזכרת לי; אם לא, הייתי משאירה את הארנק בבית.

Because ש־ means that or for the fact that. In everyday Hebrew, תודה ש... is a very common way to say thanks for... or thanks that....

So:

• תודה שהזכרת לי = Thanks for reminding me

It is more natural in Hebrew than trying to build the phrase exactly like English.

Literally, it is something like:

• ש־ = that
• הזכרת = you reminded
• לי = me / to me

So the whole phrase is that you reminded me.

The verb is from להזכיר, which usually means to remind or sometimes to mention.

Because the verb להזכיר often works with an indirect object in Hebrew:

• להזכיר למישהו משהו = to remind someone of something

So:

• הזכרת לי = you reminded me
• more literally: you reminded to me

That sounds strange in English, but it is normal in Hebrew.

If the thing being reminded is stated, you could say:

• הזכרת לי את הארנק
• הזכרת לי לקחת את הארנק

Hebrew often leaves it out if it is clear from context.

So תודה שהזכרת לי can mean:

• Thanks for reminding me
• Thanks for reminding me about it
• Thanks for reminding me to take it

The missing idea is understood from the situation. In this sentence, the next clause tells us the reminder was about the wallet.

In normal unpointed Hebrew writing, הזכרת can be either:

• you reminded (to a man)
• you reminded (to a woman)

The spelling is the same without vowel marks.

In pronunciation, they differ:

• masculine: hizkárta
• feminine: hizkárt

So the written sentence does not tell you whether the person being thanked is male or female.

Exactly. אם לא literally means if not, but in context it often means:

• otherwise
• if you hadn’t
• if that weren’t the case

Here it is short for something like:

• אם לא הזכרת לי...
• If you hadn’t reminded me...

Hebrew often leaves out the repeated part because it is obvious.

Because this is a counterfactual or unreal result: I would have left the wallet at home.

In Hebrew, a very common way to express would have... is:

• הייתי + present participle

So:

• הייתי משאירה = I would leave / I would have left
  (the context makes it clear this is past counterfactual)

This is the normal everyday Hebrew way to express this idea.

Because the speaker is female.

• הייתי משאירה = said by a woman
• הייתי משאיר = said by a man

So this sentence tells us the person saying it is feminine.

Hebrew often uses the when the possession is obvious from context, especially with personal items.

So:

• את הארנק can naturally mean the wallet
• but in context it clearly means my wallet

You could also say:

• את הארנק שלי = my wallet

But in this sentence, that would usually sound less natural or less streamlined unless you want extra emphasis.

Because הארנק is a definite direct object.

In Hebrew, definite direct objects are usually marked with את.

So:

• הארנק = the wallet
• את הארנק = marks the wallet as the direct object

English has no equivalent word here, so את is usually not translated.

Hebrew uses בית (house/home) in this kind of expression.

So:

• בבית = at home / in the house

In everyday usage, this is the normal way to say at home.

Also, in unpointed Hebrew writing, בבית can represent different vowel patterns, but here it is understood as at home.

The given word order is natural and idiomatic, but Hebrew does allow some flexibility.

This sentence:

• תודה שהזכרת לי; אם לא, הייתי משאירה את הארנק בבית.

sounds very natural as:

• Thanks for reminding me; otherwise, I would have left the wallet at home.

You could also make the hidden clause explicit:

• תודה שהזכרת לי; אם לא היית מזכיר/ה לי, הייתי משאירה את הארנק בבית.

That is more explicit, but the original is more concise and conversational.