IME, this one is hard for some protestants to grasp b/c the Catholic understanding of the economy of salvation is so different than theirs. When we “offer [something, it] up,” we offer some suffering, pain, inconvenience, etc to God and ask Him to redistribute the grace we would have received to someone else, e.g., a soul in purgatory or someone in need of conversion. We are in essence joining our suffering w/ that of Christ on the cross, thereby sanctifying the suffering.
Another way of thinking of it is that prayer takes many forms. We tend to think of it as talking to God, entering into conversation w/ Him. However, our actions are also prayers when we join them to God’s will and offer them to Him. This means that we can sanctify our labors, like the work we do at our job or our household chores. [A good reference on this is the book Holiness for Housewives.] It follows naturally then that we can offer any action or experience as a prayer, provided it is not sinful and in God’s will, of course. So, when we suffer, we offer it to God as a prayer of sorts.
When I grew up, we were always encouraged to “offer it up” for someone else; however, this need not necessarily be so. One can pray for one’s own needs as well.
I don’t know if this helps. This is a tough one to explain. A scriptural reference that may help is Colossians 1:24, which says, “Now I rejoice in my sufferings for your sake, and in my flesh i am filling up what is lacking in the afflictions of Christ on behalf of his body, which is the church …”.