Why one transistor switches on first than other
There is no need that the left transistor in your schematic will be turning ON first. I think that the article you read may be making an assumption that the left transistor is turning ON first to explain the circuit.

The setup given in the schematic is a multivibrator. The diodes in the circuit provides a positive feedback such that when ever one transistor is turned on, it pushes the other one to turn off. Its something like the output of an inverter is inverted again and fed back to the input. 

Now, when power supply is turned on, same voltage conditions appears across both the transistors and both transistors are similar (IRF540). But making exactly identical transistors is not practical hence the two transistor will have some difference in its parameters (threshold voltage, W/L ratio, etc.,). This small difference in transistor will lead to conducting of one transistor more than the other. This effect leads to reduction is drain voltage of that (fast) transistor which leads to reduction in gate voltage of the other (slow) transistor and this leads to increase in the drain voltage of the slow transistor leading to increase in the gate voltage of the fast transistor further reducing the drain voltage of the fast transistor. This positive mechanism makes the fast transistor ON and the other OFF. One can say that one of the transistor will be turning ON. Can not say which one (from the schematic provided).