Think back to when Handy warns Blossom that her past self can't see her present self, or else it would ruin the space-time continuum. This seemingly never happens when they do end up encountering each other, at worst causing past Blossom to faint. But the effect is a lot more subtle - rather than something like destroying the fabric of space and time, the time machine gets confused during their next time travel attempt because there's been a paradox in the timeline (two different iterations of the same person saw each other, which is normally impossible) and doesn't know what to do when Blossom turns the clock back. Put it this way: if there's a person from two different points in time, one of whom is time-travelling, and they meet one another, then which is the "correct" one to use as a reference in regards to set the destination time properly?
The end result is that instead of actually taking Handy and Blossom through time, it takes them through space only; the location remains persistent and largely unchanged, while the existing lifeforms are distorted into their "prehistoric" and "futuristic" versions. So for example, the Gmurfs aren't the Smurfs' actual ancestors like Handy and Blossom are led to believe, but rather approximations of what the currently living individual Smurfs would've been if they existed during the prehistoric stage instead of the modern times. All of this happens after the paradox occurs - prior to the paradox, travelling back in time doesn't seem to have any strange effects.
The only way to ultimately fix it is to travel back to around the time the clock was first used in order to fix the corruption in the timeline. Once the corruption is removed and the timeline realigns itself, anything that was done in the past gets undone, which also includes Blossom who is shown not to remember any of what had just happened. Handy, however, still has his memories (evidenced by him fainting at the end at Blossom running to him like she did when it all began) as he didn't cause a paradox by meeting himself, meaning his own space-time integrity remained intact.
