With this type of clause (see what I did there? :) ), the first thing I would do is try to reorder the words to see if I can make sense of why the "with" is there. I.e:
It's pronunciation that is the problem with Italians.
This helps you see why "with" is used.
For what it's worth, I think (generally speaking) it would be better to use "for" here, as "with" can sound a bit disrespectful.
For Italians it's pronunciation that is the problem.
(Though I don't know the full context of this single sentence.)
So, to understand the choice of "with" it is just a case of reordering the clause.
In the case of my poor attempt at humour at the start of this answer, I said:
With this type of clause the first thing I would do is try to reorder the words.
Why with? Well, try my technique of reordering the sentence to use a more typical order:
The first thing I would do with this type of clause is to try reordering the words.
The choice of "with" is harder, but as I said above, I think "for" might better". I'll explain a little bit:
The problem with
Expresses annoyance/frustration at a situation.
The problem with paying the bills is that you have to have all the money in your account.
The problem with Italians is their pronunciation (I can't understand them and that annoys me!).
Whereas "for" expresses a more neutral stance, explaining who (or what) a problem affects.
Pronunciation is a problem for Italians (unlucky for them).
The end of the month can be a problem for those with a low salary.