An early (1822) American song that uses the proverb "There's no place like home" has the lyric, "Mid pleasures and palaces though we may roam, Be it ever so humble, there's no place like home." This is exactly the sense of the cute Italian verse you provided.
The expression or sentiment "there's no place like home" seems to predate this song by many centuries, and is probably a timeless sentiment expressed in some form or another in almost all languages.
"L̶a̶ ̶c̶a̶s̶a̶ ̶t̶u̶a̶" ← WRONG
- casa tua (without definite article) = your home
- la tua casa = your house
it is not 100% true but it is pretty similar
It's just a common Italian phrase, an idiom. You can think of it like how close family members don't use a definite article (mia madre, mio fratello) -- home is so close and familiar that it drips the article. Also if I'm not mistaken "casa tua" as opposed to "la tua casa" kind of differentiates between the concept of home and house respectively.
Don't always translate literally. It will only confuse you. If you've tried translating any pages, you'll notice that literal translations often make little sense, and that a small degree of finesse is often necessary to make sentences sound right after translation. Same applies here. (House and home are often used interchangeably in many languages too.)
In languages, we don't just translate word-to-word. Such a translation usually wouldn't make any sense. We translate entire chunks of meaning at once. In many of the ways English speakers use "no", Italians also use "no". But in some cases, the sense of English "no" might better be represented by Italian "nessuno", and vice versa.
I think the meaning in English would be better if it read "like your own home"; this is consistent with most of the other comments already posted but I'm not far enough along to know if the Italian would be different if we were asked to translate "like your own home" from English. As it stands it could mean that the speaker was exclaiming how special the others person's home is and I don't think that is the intent.