//How can anyone trust and continue to vote for him? //
Put yourself in the position of an average millennial, and remember that they are effectively allowed only two choices.
On average, you're being charged something like 500% more rent than your parents were when they were in their 30s. Also, you're very likely to have entered the workforce (if you have managed to do so at all) since 2008. So, on average, your earnings have been gradually declining. Oh and your job is likely to be far, far less secure.
Also, house prices are at around 8-15 times average earnings (depending on where you are) and increasing (when your parents were your age, it was something in the region of 3-4x at a time when earnings were increasing) while your rent goes up and your earnings go down. So unless you have a wealthy family it's practically impossible for you to save up for a house, or at all (if you eventually do manage to save enough, there's a not-insignificant chance you'll be denied a mortgage because you're too old). If you're poor, the amount of state help you can get is severely limited, because housing benefit is frozen and local authorities have been ruthlessly cut.
Also because the UK has an aging population and skyrocketing pension bill, it is overwhelmingly likely that your taxes are going to substantially increase over the next few decades to pay for the enormous pension bill alone. Also you're very likely to have student debt, but that's on pretty favourable terms, so that is a plus.
Is it really so surprising that you'll support a politician who promises to change all that? Even if you don't believe he can achieve everything he promises?
Remember, we live in a two party system. All Corbyn needs to do to win voters in that situation is be more appealing than the Tories. And Tory voters - who tend to be elderly - are dying off.