For two reasons: Biden had been one of the first federal elected officials to speak on television about the attacks after several hours of media blackout, and he was Chair of the Senate Foreign Relations Committee.

Biden had hopped off the train at Union Station after taking it as was his routine (he almost never drove from Delaware to DC, and got a call about half an hour into the trip that the attacks were taking place), tried to get into the Capitol shortly after he attacks because he felt it was important that the government show it was still functioning (but was turned away by Capitol police), got word that Vice President Cheney was wanting to fly key officers of Congress to the safe cave in West Virginia (Senate President Pro Tem Robert Byrd flatly refused to leave Washington) and argued with leadership a bit about if, when, and how to do so along with trying to reassure other members of Congress and staff - along with looking for some way to get on TV for what was a combination of both publicity (never underestimate MoCs and their ability to find a camera) and what may have been a genuine attempt to try to reassure the American public.

When around noon 1 pm or so (the linked video below helped correct the written timeline here a bit) he'd finally given up hope of Congress reconvening that day and was going out to the parking lot to hitch a ride back to Delaware with a friendly Congressman, on the way to the van he was nabbed by fellow evacuee ABC chief Congressional reporter Linda Douglass, which was tremendously helpful for both of them. Douglass had been looking for any sort of senior government official to talk with, and later said that:

”It was just such a relief to see somebody of his stature and seniority, able to talk to the country, which was in a state of terror and confusion,” said Douglass. “That is the part of it that was significant to me: there were no other voices. There were no other leaders who were able to start reassuring the country.”

Besides Bush's statement following the second plane, there had been a dearth of federal officials talking, and from a reporter's perspective the Chair or Ranking Member of Foreign Relations at least could provide some idea of what the thought process was among leadership.

Here's what Biden quotes from the interview in his book:

“What I don’t want everybody to think is this is some worldwide conspiracy where there’s tens of thousands of people that are part of the army that attacked us. This is a group of people very well organized, obviously relatively well funded. And we have to figure out how their network is run. We have to penetrate it. But this is now, we can’t…just say we’re going to focus only on this kind of incident and not on chemical, not on biological, not on pathogen, not on anthrax. This is in a sense the most god-awful wake-up call we’ve ever had in terms of how we have to redirect our resources.”

“At the Pentagon,” Peter Jennings said to me from his studio in New York, “people are already saying this is so sophisticated, this has to be Osama bin Laden. Maybe so, but is the United States too focused on one man?”

“The tendency in these circumstances is to be too focused on one man, one idea, one prospect,” I answered. “I think that we should be calm. Those of us who hold high public office just calm down a little bit, collect our thoughts, collect the information in a methodical way, analyze what we know happened and what we can derive from that. I think it’s much too early for us to make those kinds of judgments. The first thing is what the president is doing. He called for calm. He’s getting in the airplane. He’s coming back to Washington, D.C., and I applaud him for that. And we should be back up and running as quickly as we can. This cannot be dealt with overnight. It’s an incredible tragedy. But it’s a new threat of the twenty-first century, and we will find a way to do it. This nation is too big, too strong, too united, too much a power in terms of our cohesion and our values to let this break us apart. And it won’t happen. It won’t happen.”

Then he hopped in the car, and outside Baltimore (so presumably around 3-3:30 pm) Bush called him to thank him for the interview. Biden being Biden, he couldn't quite keep quiet.

"I just watched you on television, he told me, and I’m really proud of you. You made us all proud. You were saying the right things."

"Thank you Mr. President for calling, I said. Mr. President, may I ask where you are?"

"I’m on Air Force One, heading to an undisclosed location in the Midwest."

When I asked him when he was heading to Washington, he said the intelligence community told him he shouldn’t.

"Mr. President, you’ve got much, much better access to intelligence, I told him, but you know that if there’s even a small percentage of a possibility of something happening, they will tell you not to come home...Mr. President, come back to Washington."

I hung up the phone, and there was silence in the van until Jimmy spoke up. “Whatever staffer suggested he call you just got fired.”

Biden was also probably at the time the single most senior elected official not in the Administration when it came to foreign policy, and even if he hadn't spoken it's likely Bush would have reached out to him at some point in the next day or two given that role, but the speech likely accelerated that.

Now, a couple caveats. Most of the recounting here comes from Biden's biography Promises to Keep with a couple bits of color from a 20th anniversary Politico article, and politicians writing about themselves are shaky to begin with as sources, especially when they're putting a book out as part of a campaign. When they write about recollections with other politicians and policy decisions it gets worse; they can be notorious for spinning things their own way, and it's one reason why you'll generally not find them footnoted much in academic books. (Bush's autobiography doesn't discuss the incident, and a very quick look through 9/11 and Bush literature doesn't turn up anything either.) When that politician is still a major player and in office, you tend to discount it further; and even at a casual glance one thing caught my eye about how Biden writes about it - it'd be very unusual to hang up on the President, since protocol generally is that he's almost always the one who decides when to end a call.

All that being said, based on Biden, Douglass, Bush's phone call of thanks, and the minutes of the day, I'd say that the general circumstances of Biden's interview were why he jumped up to the top of Bush's call list.

Edit: Found the actual interview. It starts around 1:38:13 if the link doesn't work.