I, too, used to puzzle over the idea of "giving up" one thing or another for Lent. Having been brought up within a community of folks whose sense of who they were (Baptists of an exceedingly fundamental sort) was not nearly as strong as their sense of who they weren't (Catholics), I hadn't been offered much of an explanation along the way.
More recently, having followed my heart to the East (specifically, the very Jewish-inflected early Church of Eastern Christian Orthodoxy), I've found a good bit more help in understanding the double whammy of self-deprivation and almsgiving. In that tradition, the period of Great Lent is certainly a period of fasting and self-examination, but it is no less a period of turning one's attention from oneself to others. That is to say, the fathers and mothers of the Church have constructed an efficacious ascetical program that precludes eating meat and dairy for the duration, but they have coupled that program of self-constraint with an insistence upon giving to those in need. In fact, a fast that is solely one of self-deprivation is characterized by these fathers and mothers as "a satanic fast."
Fair to say, nothing about the Orthodox way is solely a matter of turning away (from sin, bad habits, or certain foods), but is necessarily a matter of turning toward Christ. One finds, as it happens, that when one turns toward Christ, the particulars of sin, etc., are relegated to being behind him. The point here is that the energy of saying "no" to one thing or another is far less efficacious than the energy of saying "yes" to something (Someone) more desirable.
So, for those in the Orthodox Church, the season of Lent has begun. For the next eight weeks, including our Holy Week preceding Easter (what we call Great and Holy Pascha, the Passover from death to life), we are encouraged to abstain from meat and dairy in our diets, encouraged to give generously to those in need, and encouraged to take greater advantage of the life-giving, life-enhancing services of the Church, which are themselves, during this time, framed to assist our slowing down, and our moving into a more deliberate and conscious communion with the God who loves us.
One might, as well, consider this Lenten period as a period of descent. For one, it can be a period of our more frequently descending with our minds into our hearts in silent prayer, into prayer as communion with Christ. It is also a descent into our partaking of His kenosis, His emptying, His self-sacrifice that occasions our healing. Lent, therefore, becomes a salutary means of our dying to mindless habits, our dying to soul-scattering distractions, our dying to life-inhibiting illusions. It becomes a season of greater deliberation, and a recovery of our sense of the invisible Love in whom we live and move and have our being, even when we don't take notice. Great Lent is the Church's way of assisting our taking notice.
We die for a season, and then we live, live with greater awareness, and live more fully. So they say, and so I gather.