There are two aspects to honoring our “senior citizens” - legal and ethical.
The Torah says, “Rise before an elderly person, and honor the presence of a sage (Lev. 19:32).” The Code of Jewish Law defines “old” as age 70. If a 70 year-old person walks by, it is a mitzvah to stand up as a way of showing respect and honor by recognizing his presence. The obligation to show honor is not limited to standing up, but can also involve giving your seat, helping with errands, or otherwise offering assistance.
Aside from the legal mitzvah of honoring elders, there are (at least) two important ethical reasons to show them special respect. One is that they have many years of experience that we can learn from. Most people learn much from the myriad of events they experience in their lives. Especially, elderly parents and grandparents generally have specific learning experiences that they try to share with their offspring to help them be independent and successful.
But I think the most important reason to honor our elders is that each generation that precedes the following one is an essential link in the chain of our Jewish tradition that goes back to receiving the Torah at Mt. Sinai some 3,300 years ago.
Our elders are one step closer, one link closer, to that pivotal event that turned us into the Jewish nation. Each of these links is of paramount importance as it serves as the means to continue the tradition from one generation to the next. In addition, the previous generation is also one step closer than the next to the original handiwork of the Creator when God made the first people, Adam and Eve.
A true story to illustrate the importance of our elders serving as links in our tradition: In response to an anti-Semitic comment made by Daniel O'Connell in the British parliament, Benjamin Disraeli memorably defended his Jewishness with the statement, "Yes, I am a Jew, and when the ancestors of the Right Honorable Gentleman were brutal savages in an unknown island, mine were priests in the Temple of Solomon”.