'To weed something out' is an idiom meaning to remove undesirable aspects or candidates. When you weed your garden, you're removing the unwanted plants (the weeds), and leaving the desirable vegetables or flowers.
The same principle applies here to Finland's teacher education programs. They're designed to "weed out" the people who won't make good teachers. This is done early in the process, instead of waiting until prospective teachers have nearly completed training.
The author's opinion is that this is a good thing - get rid of potentially unfit teachers early on, instead of waiting until late in the process, thereby giving false encouragement to candidates who don't have the right mindset to be teachers.
Just like it's better to weed your garden when you first see the young small weeds, instead of waiting and allowing the weeds to get well established. It's more difficult to uproot the weeds if they're given a chance to establish strong roots. Plus, there's also the chance of damaging the desirable plants when you pull big weeds.