TL;DR
The majority of the Earth's iron was around when the planet formed. Calculations of the mass deposited by subsequent impacts show that these impacts were comparatively insignificant in changing Earth's iron levels, making the claim false. Additionally, those impacts came from bodies originating within the Solar System - not around other stars.
In the early Solar System, over several tens of millions of years, planetesimals collided and aggregated, forming protoplanets, which subsequently accreted even more matter (Wood et al. 2006). Many of these planetesimals formed iron cores; when they impacted the young Earth, which was molten and conducive to metal segregation, that iron sank to the center, forming a core of that was ~85% iron. This segregation ended only when silicate perovskite in the mantle crystallized. This iron core remains today, and while you can argue that the iron indeed came from outer space, it was on Earth essentially since the planet was formed, which makes the claim quite false.
To see if any large amount of iron on Earth came from space later in the planet's history, we can ask how much iron was deposited on Earth in subsequent impacts. According to the Giant Impact Hypothesis, the last major impact Earth suffered was with a protoplanet named Theia; matter thrown off by the collision subsequently formed the Moon. Assuming a reasonable value for the mass of Theia's core, and an appropriate composition, it should have deposited an insignificant amount of iron onto Earth, on the order of 1021 - 1022 kg, a couple orders of magnitude less than the iron already on Earth (Sleep 2016) (compare to Earth's mass, 5.97*1024 kg).
We could generously suggest that the originator of the YouTube video is referring to the Late Heavy Bombardment, a series of asteroid impacts that occurred about 3.9 billion years ago (Bottke & Norman 2017). A leading theory behind the LHB holds that it arose from the migration of the giant planets in what is known as the Nice model (Gomes et al. 2005), perturbing a relic population of asteroids. This, too, could not have deposited much iron on Earth (Ryder 2002), judging by impact rates on the Moon.
Interstellar interlopers are likely not a significant source of iron, given the local density of 'Oumuamua-like objects (Do et al. 2018). With only 4 Earth masses worth of interstellar minor planets per cubic parsec (according to Do et al.; Engelhardt et al. 2017 give a value three orders of magnitude lower), it's unlikely that Earth has experienced any collisions with any over the course of its existence, which should put to rest the claim that iron came from meteorites from other stars.
The above show that impacts were not significant sources of terrestrial iron - although at any rate, this point is moot, since we've already shown that there was iron when Earth formed, and any amount of iron on the planet would invalidate the claim.
As an aside, I'd like to specifically address something mentioned in Vincent's answer:
There might be iron produced on Earth by the radioactive decay of e.g. uranium, but this should be a very small quantity compared to the amount present when Earth was formed.
The main contributors to radiogenic heat inside Earth are uranium-238, uranium-235, thorium-232, and potassium-40 (Korenaga 2011); most models of radiogenic heat assume that all radiogenic heat on Earth comes from those four isotopes. As the latter has an atomic mass less than iron, it clearly cannot decay to iron. We can look at the decay chains of the other three elements (uranium-238, uranium-235, thorium-235) and see that none of them produce iron. Therefore, as there are no major natural fusion pathways operating on Earth to form iron from lighter elements, it seems that Earth is not producing a significant amount of iron from any nuclear process, including fission.
