Start of year 1, with $100 BV. Let's say earnings are $7 by end of year 1 and BV now is $107. Why should it be worth more than 1x BV in this case?

Because cash doesn't just stack up in operating companies. (outside Japan)
It typically gets used for expansion capex, which earns a decent return on average.
Absent other information about management being dumb, let's assume it earns the same rate of return as the existing assets.

So, in round numbers, if ROE isn't falling and assets are climbing, the firm is worth more.
Let's assume it's a big company with a serious going concern of a business.
If you value it on earnings power value (as one should for earnings assets) rather than asset value, each dollar retained is worth more than a dollar.
A typical solid firm gets a pretty good return on its assets. Median among S&P 500 firms lately is 16.7%.

In your example, they had $100 of book in year one. If they are like a current median big firm, they are earning 16.7%, so $16.70 net profit.
In year two they had $107 of assets in year two. Since typical ROE is high, that was a bad year, but let's go with it.
But year two is typical for returns and they earn the currently typical l6.7% on assets, or $17.87 on $107 in assets.
The difference is $1.17/year. Would you pay more than $7 for earnings of $1.17? I would.
If so, then the $7 in retained book is a worth a multiple considerably greater than 1.

Operating companies typically trade at P/B ratios well above 1.
In short, the reason is that their situation lets them allocate that capital at higher returns than you're likely to get.
So an incremental $1 held and managed by them is on average worth more than $1 to you.

Jim