When an alpha particle speeds through air it rips electrons off the molecules that it passes close to. The electrons don't just attach themselves to the alpha particle and neutralize it. They're flung a long way away from their parent atom and the air ion that's left tends to stick around for some time afterwards.
Every time it ionizes an air molecule the alpha particle loses a little energy to the emitted electron and slows down. After a few thousand ionizations it has almost stopped and the alpha grabs a couple of electrons from nearby molecules and ends up as a neutral helium atom. Perhaps counter-intuitively the alpha makes more ionizations per distance when it's going slowly than when it's going fast. If you imagine that the electron has to reach a certain speed before it is ejected from the atom then the faster the alpha moves the less time there is to reach this speed before the alpha is out of electrostatic range.
Beta particles have a much smaller mass than alpha particles. This means a beta with the same kinetic energy as an alpha will be moving much faster. Because it's moving faster and also has only half the electric charge it is less likely to ionize an air molecule as it whizzes by. For this reason the ionizations from a beta particle are much more spread out than for an alpha. An alpha and beta of the same energy will make similar numbers of ionizations before stopping. But the alpha particle will make them in a shorter distance. This is what we really mean when we say alpha radiation is more ionizing than beta.
Betas are moving faster but are close to 8000 x less massive. Alpha ionisation take place over a single cell if incident on biological matter and a few cm in air, so more properly one ought to say that the number of ionising events per mm of path is greater for alpha than for beta, particularly at the extremity of their ranges. Furthermore, because betas exhibit range straggling, being emitted with a range of initial kinetic energies, and since they will ionise only on direct interaction the early secondary electrons emitted will also have ionisation potential. Ballpark numbers from bubble chamber tracks are alphas 10^5 and betas 10^2 (max) ion-pairs per mm.
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