The Romans themselves were an Italic tribe. They conquered the others but much of that culture was mixed and assimilated into Roman culture. Just as an offhand example the Roman name Pompeius is actually Oscan rather than Latin and is the equivalent of Latins' Quintus (from the Latin word for five: 'quinque', in Oscan 'pompe'; Latin 'qu' generally becomes 'p' in Oscan; 'quod' -> 'pod')

Anwyay, the other Italic languages survived Roman conquest as regional languages for centuries after. Not least Oscan, which persisted longer than e.g. Umbrian, until to the first century CE. It'd used a variant of the Old Italic script, although that may have stopped a bit earlier in the first century BC (although an end to the production of new inscriptions does not mean knowledge of how to read it was lost). A decent amount is known about this language because they left behind a great deal of inscriptions down in Italy.

It was down there in Italy where Germanic peoples from up north found them. We know they traveled there to serve as mercenaries in the Roman armies, and it's possible purely trading journeys were made as well, but in any case the result is that from the first century CE we start to see significant cultural changes in north Europe due to Roman influence and you start to find Roman artifacts from that time as well - coins, art, glass, but also influences on farming implements and other practical things. (one fellow from Skivarp in Sweden was buried with a whole Roman dinner set) In 98 CE you have Tacitus writing Germania as well, the first ethnographic work about Germanic peoples. So there is ample evidence of the broad cultural exchange going on and that northern Europeans started emulating Romans in the same era as the runic alphabet is introduced. So that is one major reason it's believed to come from Old Italic.

However it's a bit of an exaggeration to say runes were 'based' on Old Italic script; a better term would perhaps be to say it's inspired by it. It's not generally believed there was any attempt to directly fit or adapt the Old Italic script to their language. The idea (or one idea; there are competing theoires) there would be that they borrowed the concept and created their own new alphabet while borrowing a few letters (l, s, o) with sounds from Old Italic as well as the forms of some others, but also a few from Latin (r, b). So if this hypothesis is true, they wouldn't really need a very deep or detailed knowledge of the Old Italic alphabets anyway.

It's by no means settled it came from there though; but regardless of where the shapes came from, a telling sign it was a new invention is that the order is entirely different; on the whole alphabets have alphabetical order (a-b-c/g-d-...) and it's (on the whole) remarkably conserved from Phoenician to Hebrew to Greek to Cyrillic to Latin. But runes had futhark (f-u-þ-a/o-r-k) order rather than this. So it's a quite a noteworthy thing that a script which must have have been inspired by some alphabetic script does not have this feature which is otherwise highly conserved.

The Latin alphabet as it was at the time (e.g. on the Arch of Titus) also not so suitable for a script that would primarily be cut into wood. Wood was always the main medium for runes. (although for conservation reasons preserved wooden runic inscriptions get less common the farther back we go) A 'good' script for that would have only straight lines - curves are harder to make, and not too many either 3-4 per character at most. It should not have fine angles that could be difficult to distinguish; preferably only 90 and 45 degree angles. And last but not least a good carving script should not have any horizontal lines, because lines parallel to the wood grain are very difficult to see. The runic script obeys these "design criteria" but the Latin one doesn't do so as much as the Old Italic ones did. So another possibility is that, seeing several alphabets in use in the Roman world, chose to take inspiration more from an older variant more suited to their purposes.

But the question is far from settled. Henrik Williams, who's a professor of runology at Uppsala University proposed a derivation from the Latin alphabet purely on these kinds of visual design criteria while completely disregarding sound values. (unlike the Old Italic idea stated earlier) Williams adds another criteria that runes obey - that lines on runes never go above or below the vertical line/staff. Which is true but it's not obvious why it'd be an important criterion.

In summary we don't know exactly when the runic alphabet was created; it's a matter of how long one is prepared to believe it was around before leaving an inscription we'd find. It could've been as early as the first century BC, when some Old Italic scripts were still being used. In any case though, the creators of the runic alphabet did not necessarily know that much about Old Italic script. And that's all assuming it really was based off Old Italic and not Latin as Williams argues. Nor Greek or some of the less popular suggestions that've been made and which we've no doubt not seen the last of.