I get why the 'missing link' question comes up, it's a common one, and evolution can seem confusing at first. But let's break it down simply, like we're chatting over coffee.
Imagine evolution not as a ladder or chain where one animal turns directly into another, but as a big family tree. You don't evolve from your cousin, you both share a grandparent. Most fossils we find are like distant cousins or aunts/uncles, not your direct grandma. We have tons of these 'relatives' in the fossil record showing how groups changed over time, but we don't need every single one to see the family resemblance. Especially with DNA nowadays that can show the connection even more clearly.
Think of the fossil record like an old family photo album with some pages missing or photos lost. You might not have a picture of every birthday, but with the ones you have, you can still see how the kids grew up, changed hairstyles, and became adults. We have thousands of 'photos' (fossils) showing gradual changes, like Tiktaalik (fish to land animal), Archaeopteryx (dinosaur to bird), or early human relatives like Australopithecus. Each new find fills in details, but it doesn't 'break' the story if some are missing.
Fun twist: Every time we find a new fossil, it creates two new 'gaps' on either side, like splitting a puzzle piece. But that doesn't mean the puzzle is broken, it just gets more detailed.
You're right that we don't see big changes in one lifetime, this is because evolution takes millions of years, like watching a glacier move. But fossils show it happened, and we see small changes today (like bacteria evolving antibiotic resistance or finches adapting beaks in real time).
Evolution isn't about one species suddenly becoming another, it's tiny changes adding up over generations, like compound interest in a bank account. We see it happening now in labs and nature (e.g., new virus strains or insects resisting pesticides). As for probability, it's like saying winning the lottery is impossible, yet people win because there are billions of 'tickets' (mutations) over deep time.