What does "I am married with three kids" mean?

If you're talking about you and your spouse, it's not possible to be "married with" anyone; you can only be "married to" someone.

Hence there is no ambiguity in saying "married with three kids", as the "with" cannot be associated with "married".

Did you hear it or read it?

If you heard it, you may not have correctly heard the comma, which de-nastys to:

I am married, with three kids

It means the speaker is married, and has three children (generally living with them as dependents).

Perhaps it isn't the best grammar. It certainly isn't the clearest way of putting it. However, it references the IRS (USA tax) code, in particular someone who is filing a joint return with a spouse and has other minor dependents. Several examples of this construction can be seen here. As such, any person who fills out a USA tax form is liable to know immediately what it means.

This terminology also was used in the name of a hit TV series.

I am married with three kids.

can safely be resolved to

I am married, with three kids.

at which point the context & the meaning become completely clear. I don't think the sentence sounds nasty, as nobody would ever want to imply what you are referring to.

Great! As a non-native English speaker from another place of the world, sometimes you just don't figure out some of the rules for translation-that should obey the side of the foreigner language and not your own. This case is a good example.

In Spanish you say: "Estoy casado con Tania".
The wrong translation is: "I'm married with Tania".
The OK translation is: "I'm married to Tania".

In the question there's obviously a mistake which has its origin in a wrong translation. Clearly, the punctuation helps to give a satisfying connotation, but the problem isn't there.

"I'm married to three children" is what "Estoy casado con 3 niños" means in Spanish and is totally wrong, mainly because the word con means with most of the time. In this case, the translation should become not word by word but understanding the meaning of the phrase. Much of the time the Spanish speaking people fall into this kind of mistakes out of the logic that Spanish language has implicitly, but to learn English as well as other idioms, you most open your mind to break those language barriers.
"I'm married with three children" is "Estoy casado con 3 niños", nothing wrong, but "I'm married to three children" is the nasty meaning you were seeing.

In case of a translation, the fix for it would be to add the comma. Simple.

Anyway, this has to do with a wrong conception of the verbal form and how it is used.
My two cents, I'm new here and love idioms. I'm always learning something new and this is an excellent place for doing that.