The economic reasons I can think of largely involve more rapid transportation between hubs all over Asia (and maybe even Europe) to hubs in North America. A trip across the Pacific from Hong Kong to Seattle can take two or more weeks, while a rail trip from Hong Kong to Seattle could be done in perhaps one week, depending on how many yard changes would be needed. (Transit times between Hong Kong and the East Coast via the Panama Canal are even longer, taking a month or more, while the additional time required to cross Canada or the US would be measured in days.) Using Google Earth and some admittedly straight lines, the distance from Hong Kong to Seattle was about 6600 miles. If a train can average even 60MPH over that, the trip would take less than five days, and even some curves and detours wouldn't extend it by much. Of course, most train traffic wouldn't originate from Hong Kong, but would instead go directly, more or less, from the other hub cities scattered across China, reducing the factory-to-destination time even further.
Rail gauges might not even need to be considered, since the US and China use the same gauge, and the tracks through Siberia could be laid as dual-gauge or even just 1435mm gauge and the Russians can start adopting that (it would make trade with Europe easier, too).
Such a bridge would have to allow a significant amount of rail traffic to cross, but the economics could work out over a very long term (many decades at least). The trillion-dollar price tag is for a network of roads and rail running from London to New York; the bridge itself would probably be in the range of $100 billion for a road and dual tracks. Amortizing that at 2% interest over 50 years gets annual costs of $3.18 billion for the loan itself.
A North Carolina Dept. of Transportation study placed the approximate cost of a 4000 SEU Panamax vessel at 80% capacity at about $1500 per TEU and a New Panamax (capacity 12,000 TEU) at 51% capacity at about $950 per TEU. Those capacities can be matched using 4.5 or 8.5 trains, respectively, of 180 wagons (the max length allowed in the US) double-stacked and able to handle four TEU each (so 720 TEU). I'm not sure about the basic economics, but I imagine that the costs for train travel are less than that. Even if they're higher per day, they would probably be lower per trip.
If the toll per TEU is about the same as it is in Panama ($72), each nearly-full train crossing would bring in about $50,000. If maintenance consumed a quarter of that and the rest went to the loan, it would require almost 85,000 annual train trips, or about 232 per day. Even at zero interest, it would require more than 53,000 annual train crossings, or about 146 per day, and all of those at around 95% capacity.
However, if the tolls were higher but the cost per TEU were lower, it might work out. At 50 trains per day, the toll would need to be about $250 per TEU (plus some amount for maintenance) to pay off the loan. That's still a lot of trains for two tracks, but it might be workable. This doesn't include any road tolls or oil/gas transit fees for lines running along the bridge, which could add a fair amount, but I'm not sure it would dent it significantly.
Another reason that I can think of, though, is to get part of North America reliant on Russian natural gas, particularly as Alaska's petroleum-derived production slows over the coming decades. That could bring an influence level that's hard to achieve any other way. Russia has a history of slowing or shutting off gas supplies to Ukraine and other places during winter when it wants leverage. I'm sure it would love to have that leverage over the US and Canada as well.