To give the gist of this passage (and the bits before and after it), Thoreau is arguing against the concept of "wisdom [that] comes with age", or "the wisdom of the elders".
In Thoreau's view, most people have dedicated their lives to unimportant goals, e.g. accumulating wealth.
By a seeming fate, commonly called necessity, [men] are employed, as it says in an old book, laying up treasures which moth and rust will corrupt and thieves break through and steal.1 It is a fool's life, as they will find when they get to the end of it, if not before.
Besides being dull, these goals are also difficult or impossible to fully satisfy, leading these people to judge themselves as failures even by their own standards. Hence, Thoreau's famous assertion that "[t]he mass of men lead lives of quiet desperation".
Because these people cannot imagine an alternative, meaningful way of living—because they "honestly think there is no choice left"—they do not (in Thoreau's opinion) learn anything useful from their experiences. They spend their entire lives barking up the wrong tree. Thus, age does not bring wisdom.
Practically, the old have no very important advice to give the young, their own experience has been so partial, and their lives have been such miserable failures... I have lived some thirty years on this planet, and I have yet to hear the first syllable of valuable or even earnest advice from my seniors. They have told me nothing, and probably cannot tell me any thing, to the purpose. [...] If I have any experience which I think valuable, I am sure to reflect that this my Mentors said nothing about.
If we accept the premise that our elders and ancestors gave us a poor example of how to live life, and that we shouldn't follow in their footsteps, then the only logical alternative (which is what Thoreau proposes) is to blaze a new trail and do the things they didn't do.
What old people say you cannot do, you try and find that you can. Old deeds for old people, and new deeds for new.
When Thoreau contrasts "old people" with "new [people]", he doesn't simply mean seventy-year-olds versus twenty-year-olds, although there is an element of that. The "new [people]" are, in Thoreau's vision, a new kind of people, making a clean break from a species of "tedium and ennui" that is "as old as Adam".
...man's capacities have never been measured; nor are we to judge of what he can do by any precedents, so little has been tried.
In short, Thoreau is calling for a kind of spiritual progress.
As an aside, the phrasing of the bolded sentence reminds me somewhat of that biblical metaphor about putting new wine into new wineskins (or "bottles", in the King James Version). This might be intentional, since Thoreau makes other biblical allusions in the text, but I could be reading too much into it. Judge for yourself:
Neither do men put new wine into old bottles: else the bottles break, and the wine runneth out, and the bottles perish: but they put new wine into new bottles, and both are preserved.
And no man putteth new wine into old bottles: else the new wine doth burst the bottles, and the wine is spilled, and the bottles will be marred: but new wine must be put into new bottles.
1 Matthew 6:19