To właśnie jest mój problem.

Here to is a demonstrative word meaning something like this or that.

In To właśnie jest mój problem, to points to a situation, issue, or thing already being discussed:

• To jest mój problem = This/That is my problem
• To właśnie jest mój problem = This/That is exactly my problem

A common mistake for English speakers is to think of to as the infinitive marker to in English, but it has nothing to do with that here.

It can mean either this or that, depending on context.

Polish to is often less specific than English this/that. English forces you to choose, but Polish often does not. So this sentence could be understood as:

• This is exactly my problem
• That is exactly my problem

The surrounding conversation tells you which one sounds more natural in English.

Here właśnie means exactly, precisely, or that very thing. It adds emphasis.

So:

• To jest mój problem = This/That is my problem
• To właśnie jest mój problem = This/That is exactly my problem

In other contexts, właśnie can also mean just now or right now, but that is not the meaning here.

Both are possible, but the emphasis is slightly different.

• To właśnie jest mój problem puts emphasis earlier on właśnie, so it strongly highlights that exact thing
• To jest właśnie mój problem is also natural and may sound a bit more neutral in some contexts

Polish word order is more flexible than English word order. Moving words around often changes emphasis rather than basic meaning.

Because problem is a masculine singular noun, and mój has to agree with it.

Possessive adjectives in Polish change form depending on the gender, number, and case of the noun:

• mój problem = my problem (masculine singular)
• moja książka = my book (feminine singular)
• moje auto = my car (neuter singular)
• moje problemy = my problems (plural, non-masculine-personal)

So mój is the correct form for problem.

Because in this kind of to jest ... identification sentence, Polish normally uses the nominative.

So you get:

• To jest mój problem
  not
• To jest moim problemem in the usual neutral sense

Compare:

• Jan jest lekarzem = Jan is a doctor
  
  • here the predicate noun is often instrumental
• To jest lekarz = This is a doctor
  
  • here it is nominative

Your sentence is the second type: a this/that is X structure, so problem stays in the nominative.

Because Polish has no articles.

English uses a/an and the, but Polish does not. So Polish simply says:

• mój problem

and context tells you whether English should say:

• my problem
• the problem
• sometimes even a problem, depending on the sentence

In this case, mój problem naturally means my problem, and English may add emphasis such as exactly my problem.

Because ten usually goes with a noun, while to can stand by itself.

For example:

• ten problem = this problem
• to jest mój problem = this/that is my problem

So if you want to point to something without immediately naming a noun, to is the normal choice.

In full, standard Polish, jest is normal here:

• To właśnie jest mój problem

In casual speech, people sometimes drop jest in certain situations, especially in informal or emotional speech, but for a learner, the full version with jest is the safest and most natural form to use.

So if you are learning, stick with:

• To właśnie jest mój problem

A rough English-friendly approximation is VWASH-nye.

A few useful points:

• ł sounds like English w
• ś is a soft sound, somewhat like a softer sh
• nie at the end sounds roughly like nye

So the whole sentence is approximately:

• To VWASH-nye yest mooy PRO-blem

This is only an approximation, but it can help at the beginning.