This is insanely the best question ever. People will say and probably have said, it means you just bind your problems to Christ on the Cross, or leave your problems etc at the Altar…
An then exactly how does one do that ? It just becomes a metaphor at some point for prayer, there is no physical way to just mysticly offer up ones sins, problems, joys, what have you, in the form of incense or a lit candle , or ringing a bell, or whatever, we cant go to the physical cross and tie anything to it, all one can do is pray and hope God answers one prayer in the physical time frame that it would be helpful, not 10 years down the road and then a light bulb comes on an one is sittn where ever and goes oh wow, yeah i had this problem and it would have been great had it been solved 10 years ago so that me an my family didnt have this giant falling out and now we dont talk to each other but at least my drug habit is gone, or something to that effect, just an analogy.
So the only rational answer i can come up with is, offering it up means, praying. For whatever.