Q: However we understand how prayer works for ourselves (whether along the lines of
Rav Yosef Albo or not), how can we understand the point of praying for someone else?
How can my prayer help someone else?

A: It is important to begin by taking note of the fact that the ideal prayer is that of the
person in need. On one level, the answer to the question is that prayer really does
belong to the person in need. Chazal teach [Bereshit Rabbah (53:17)] that the prayer of
a sick person on his own behalf is the most effective of all.
  At the same time, there is a concept of prayer on behalf of others, and it needs to be
understood. The Sheiltot has a remarkable understanding that sheds tremendous light
on this issue. The Gemarah in Rosh Hashanah (17b) distinguishes between prayer
nullifying a decree against an individual and against the community. The Sheiltot (no.
66) uses Moshe’s prayer on behalf of the Jewish People after the Golden Calf as an
example of a nullification of a decree against an individual! The Netziv (ad loc. §10)
points out that according to the Sheiltot, the status of individual/ community is
dependent on the prayer, not the decree/ punishment, per see. How are we to
understand this?
  Apparently, the point of the Sheiltot is the following: Moshe can not pray on behalf of
Benei Yisrael in terms of the consequences for them. Only they can. This is in line with
chazal’s point that prayer fundamentally belongs to the person whose need it is.
Nonetheless, Moshe can pray about the decree against them, in so far as he is affected
by it. That is to say, he can’t pray on their behalf; but he can pray on his own behalf. To
the degree that their fate affects his own, he is able to pray. Harm to the Jewish People
hurts Moshe, therefore he can pray for them. The decree in question is therefore a
decree against an individual, not a community, in terms of his prayer.
  This is the secret of prayer on another’s behalf. We can pray for someone else
because the fate of those we care about affects us. To the degree that we care about
other people, and only to that degree, we are able to beseech God on their behalf. The
greater the measure of our connection, the greater our empathy for their plight, the more
effective is our prayer. Our degree of v’ahvta lereacha kamocha determines how close
we are able to come to the ideal prayer of the choleh himself!