How did New Hampshire and Iowa become so important in the US Presidential electoral process?

For Iowa. It became the first in the nation in 1972.

A lot of Democrats found the turmoil that surrounded the 1968 convention to be distasteful. So in Iowa, along with other states, a movement was started to allow more public input into the selection of delegates. Iowa decided to hold caucuses which would select the Iowa delegates for county, district, state, and national conventions. This would allow direct public input in the first round, and a lot of public access to the delegates. Because they decided to create a process with caucuses followed by four rounds of conventions, the caucuses had to be held early. But according to Cliff Larsen, the Iowa Democratic Party Chair, there was no intention of trying to be first in the country per se. They did however realize it would be early.

Then the Republicans decided to match the Democrats dates and process. To everyone's surprise the national media showed up to the caucuses. The Republicans added a straw poll to their caucuses, to give the media numbers to include in their stories. And suddenly Iowa, a state that was used to being ignored in national elections, was suddenly being courted by national candidates and media.

This gave Iowa some nice perks, they got a real voice in national politics for the first time. And it also worked for the candidates. Instead of working the whole country at the same time, it let candidates focus on a single state. It let a candidate like Jimmy Carter create a narrative that he could win in the Midwest was well as the South.

So as time passed it became a tradition that Iowa fiercely defended. It may behave been serendipitous that they ended up at the start of the political calendar, but they have fought hard to stay there ever since.