Based on the story's context, William simply felt that he couldn't bear to be apart from Phillipa, even in death. In all the years of their marriage, "they were never apart for more than a few hours." In fact, the couple had been together since the day they jointly won the Charles Oldham Memorial Prize.
Also, both Phillipa and William had been naturally competitive with each other before Phillipa's death. There is every indication that William wanted to best Phillipa, even after her death: "Forgive me, but I had to let her know." What this quote means is that William wanted to let Phillipa know there was such a word as "whym-wham" in the Oxford English Dictionary. He decided to follow her into the grave (and presumably into the afterlife) in order to let her know this.
On the surface, this might seem a little extreme. Why would a man kill himself simply to prove the existence of an obscure word? To answer the question, we must remember that William had always enjoyed sparring with Phillipa while she lived. From the moment he fell in love with her, his affection for Phillipa had been wrapped up in a friendly rivalry. The narrative implies that William would be lost without Phillipa and that he would find it unbearable to live without the pleasure of the good-natured rivalry they had always enjoyed.
The story of why William took his life actually began with a crossword puzzle. For many mornings, Phillipa had relished completing the Times crossword before William got to the breakfast table. On one particular morning (to William's great amusement), Phillipa failed to complete an entry. So, based on the clue "Skelton reported that this landed in the soup," William filled in the word "whym-wham" for the last entry. Although Phillipa argued against the existence of such a word, William was adamant that the word existed.
Phillipa died of a heart attack before William could win his bet. By the time the Senior Tutor and Master at Merton College (one of Oxford University's 38 colleges) knocked on William's door, the elderly professor was already dead from a self-inflicted gunshot wound. William preferred to die rather than live without his beloved Phillipa.