Mitochondrial DNA is passed from mothers to offspring. Fathers don't, as a rule, contribute this kind of genetic information to the next generation. Currently, scientists don't know for certain what the evolutionary advantage is for this arrangement. But they assume there is an advantage, because the pattern is so widespread: It must have evolved for a reason. One speculation is that the genetic information passed through the male line tends to get damaged just prior to fertilization. Therefore -- according to this theory -- organisms are better off jettisoning it.
To better under all this, let's review the basics.
In the cells of animals and other eukaryotes, DNA is found in the nucleus. But the mitochondria are distinctive among the remaining organelles: They, too, carry DNA -- genetic information that regulates their own functioning.
Both egg cells and sperm cells contain nuclear DNA, each bearing half the complement of chromosomes of a normal, non-gamete cell. During fertilization, the nuclear DNA from each gamete comes together. The resulting offspring gets nuclear DNA from both parents.
For mitochondrial DNA ("mtDNA"), it's different. Both eggs and sperm have mitochondria. Therefore, they both carry mtDNA. In fact, the sperm need their mitochondria to work very hard -- they require lots of energy to swim and compete with other sperm.
But somehow, during fertilization, the sperm's mtDNA usually disappears. How? In some species, it appears that the sperm's mitochondria get actively eliminated by the embryo (via the action of autophagosomes). But in 2016, scientists studying fertilization in the roundworm, C. elegans, discovered that the sperm of this species have a kind of
"...internal self-destruct mechanism that gets activated when a sperm fuses with an egg."
The sperm's mitochondria began to break down before the autophagosomes came along.
We don't yet know how the sperm's mtDNA are eliminated in humans. But it's clear that this happens, and while a little male mtDNA sometimes slips through, matrilineal inheritance is the rule in most multicellular organisms. Mitochondrial DNA is usually passed down from the mother only.
Why do we see this pattern? It appears that both sperm and embryo make an effort to destroy paternal mitochondria. Usually, organisms don't engage in costly behavior without a payoff. So biologists assume there is a benefit.
In the experiments on C. elegans, the scientists found that delaying the destruction of paternal mitochondria reduced the changes that the developing embryo would survive. As a researcher comments in a NY Times article (see link below), this evidence supports the idea that paternal mtDNA is knocked out in order to prevent health problems:
"This paper provides the first experimental data suggesting that it’s not good to keep sperm mitochondrial DNA."
Possibly, the paternal mtDNA is at high risk for becoming damaged because of the very high demands placed on a sperm's mitochondria. They exert a lot of energy while they are swimming and jockeying among other sperm in order to fertilize the egg, and some researchers speculate that mtDNA could get damaged or suffer mutations as a result.
If this speculation is correct, then the evolutionary advantage of matrilineal inheritance is that it protects individuals from acquiring deleterious, damaged mtDNA.