What does homie mean?

Homie came from WW2 Chicano veterans; they would call other Chicanos Homes, Homeboy, Homie. Like the little homie characters.
So yes, it is a Latino slang.

As far as etymology, homie is short for homeboy or homegirl. A homeboy or homegirl is a close friend with whom you have a lot of shared history or similar experience. They do not have to necessarily come from the same hometown or neighborhood. And the length of time of your familiarity is unimportant. Being around and sharing experiences with the same person everyday for the majority of the day will breed enough familiarity between each person for them to be homies in very little time, versus someone who you’ve known your entire life but share nothing or have nothing in common.

Conversely, two strangers could be homies if they are both from near identical backgrounds, upbringings, ideologies, affiliations, etc. You might address someone as homie if you wish to start a relationship based on those similarities. But, you would not introduce someone to a third party as your homie unless a level of trust between you and the person you are introducing has been previously established.

As Andrew has pointed out, it can be used in a derogatory manner when used to mimic, mock or imitate a person from the culture or vernacular from which the term originated. Much in the same way any accent or dialect can be used to mock someone with that particular accent or vernacular. It can also be used in a threatening or aggressive manner. Much the same way as the word “friend” can be used with a threatening or aggressive tone to one’s voice when addressing someone who is clearly not your friend.

Not too many years ago I would have advised you against using the term "homie", as it is connected to the vernacular of a particular ethnic group, and you wouldn't want to sound like you were mocking them by using it incorrectly. Nowadays the word is much more common, so, given that Dicaprio can use it without causing offense, you might sound out-of-touch, but not actually offend anyone.

Hopefully. It can still be used in a racist manner, so you should still use good judgement.

The etymology of "homie" is unclear, but since it's most commonly associated with Latino slang I always assumed that it derived from the Spanish hombre, which means "man" but is used to refer to any male who you are close to or at least friendly with. Wikipedia gives other possible explanations.

The meaning is similar to saying someone is a "buddy", but the actual nature of the relationship can vary from person to person. Dicaprio doesn't literally mean that Winslet is from his home town (he is from California and she is from England) so his meaning is clearly that she is a "close friend". Meanwhile, someone else might use "homie" to mean "someone I would trust with my life".

Meanwhile, someone else might use "homie" in a scornful way, to indicate animosity. You will just have to judge from context.