Advocates for a "living Constitution" say that things have changed since the Constitution was adopted way back in 1789 and that judges [usually the Supremes] should change the way the Constitution works "in order to bring things up to date". The appeal to the courts is a way of side stepping the democratic requirement that laws are only made or changed by the duly elected legislature. It's easier to sell your changes to 5 out of 4 unelected lawyers possessed of lifetime tenure than it is to sell them to the much larger legislature all of whom needs to stand for reelection in the near future.
The proper way to change the law in a democracy is to have the votes to pass your changes into law in the elected legislature. Many will complain that this is just too difficult, which is another way of saying that they don't have the votes for their pet programs. The Constitution allows for amendments. We have made twenty seven amendments since 1789, the most recent in 1992. It can be done, but the Constitution calls for super majorities in both houses of Congress and among all the states. Amendments only happen with widespread political support.
The late Justice Scalia was opposed to the notion of a "living Constitution". He preferred to call it the enduring Constitution. I'm with Justice Scalia in this.
Obama wants a "living Constitution" person to replace Justice Scalia.