I'm relatively new to D&D and was wondering how the second part of the witch bolt spell in 5e works. The first cast allows me to recast it until the target exits the spell range, moves into full cover, or I cast something else.

I understand it takes one spell slot to cast, but does re-casting it also use up a spell slot?

No. It takes only the initial spell slot.

The spell is cast only once, and as an effect of the spell, the additional bolts occur on subsequent turns. These do not count as additional 'castings' of Witch Bolt.

Witch Bolt has a duration of 1 minute. The text says:

... Make a ranged spell attack against that creature. On a hit, the
target takes 1d12 lightning damage, and on each of your turns for the
duration, you can use your action to deal 1d12 lightning damage to the
target automatically.

The text in bold does not say 'you can use your action to cast the spell again' only 'you can use your action to automatically deal damage. This does not cost you any more spell slots (but you can't cast any more action-Casting Time spells while you sustain this effect)

The Action provided by Witch Bolt is not "Recasting" the spell

The first cast allows me to recast it until the target exits the spell range, moves into full cover, or I cast something else.

Emphasis mine, is where you're confused. No part of Witch Bolt involves a "recasting" of the spell.

Let's reread the description for Witch Bolt:

A beam of crackling, blue energy lances out toward a creature within range, forming a sustained arc of lightning between you and the target. Make a ranged spell attack against that creature. On a hit, the target takes 1d12 lightning damage, and on each of your turns for the duration, you can use your action to deal 1d12 lightning damage to the target automatically. The spell ends if you use your action to do anything else. The spell also ends if the target is ever outside the spell's range or if it has total cover from you.

Also emphasis mine.

What Witch Bolt allows you to do is use your action repeatedly on subsequent turns to reapply the damage, without needing to make another Attack Roll or force another Saving Throw. So long as the target remains in range, you have a guaranteed damage source for the duration. That's the strength of this spell. Nothing in the description says anything about "recasting" the spell; this is simply an Action used to invoke this damage. The only time you'll cast the spell again is if the duration ends (naturally or early) and you decide to cast the spell again.