Can someone explain this result? (Rule of three)

Solution 1:

It takes $1$ machine working alone $210 \times 6= 1260$ hours, not the $210/6 = 35$ hours in your second attempt

So it takes $10$ machines working together $1260/10=126$ hours, as in your first attempt

Solution 2:

$\frac {210}6$ does not have any meaning in this context. The important thing is that the work takes $6 \cdot 210=1260$ machine-hours. You can divide that by the number of machines to get the number of hours needed, which gives $210$ for $6$ machines and $126$ for $10$ machines. It is often helpful to write out the units. What would hours/machine (which is what $\frac {210}6$ gives) mean? If the machines worked one after another on the same item it would be meaningful as the amount of time each machine takes.