The meaning of today's phrase is simple: non ti preoccupare, 'don't worry about it'.
Non ti preoccupare, ci penso io.
Don't worry about it, I'll take care of it.
– Scusa il disturbo.
– Vai tranquillo, non ti preoccupare.
– Sorry to bother you.
– Don't worry about it, go ahead.
While preoccupare comes from the same Latin root that gave us 'preoccupy' in English, this Italian verb means 'to worry' or 'to fret'. Just imagine your head being so 'occupied' by anxieties that you can't think of anything else.
But you don't need to concern yourself with that, since we're talking about doing just the opposite.
The only thing you might need to (ahem) worry about is whether you should say non ti preoccupare or non preoccuparti, which is such a source of confusion that even Italians have dedicated heated forum threads to it.
The short answer is: you can say either interchangeably.
They both mean exactly the same thing, non preoccuparti is just the reflexive form of the verb. It's the difference between 'worrying' and 'worrying yourself'.
Just bear in mind that preoccupare is transitive (it needs a direct object), so you use it to say that something worries you. In this case the verb agrees with the thing or things (third person singular/plural), not with the person it's worrying.
La sua salute mi preoccupa.
Her health worries me.
Preoccuparsi, on the other hand, is intransitive (it doesn't need a direct object) and you use the preposition per to specify who or what you're worrying about. The verb is conjugated according to the person doing the worrying.
Mi preoccupo per la sua salute.
I worry about her health.
If it helps you decide between the two, according to Google there are around 5 million more hits for non preoccuparti than non ti preoccupare.
But personally I prefer non ti preoccupare – or as we say in Rome, non ti preoccupa'.
Don't spend too much time thinking about it: just pick one and… you know the rest.
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