Why are metalloids found in a zig-zag line on the periodic table
In addition to Pritt Balagopal's answer, in which he has clarified the trends in metallic character across the Periodic Table, I would also like to explain why the metalloids occur in a zig-zag fashion. 

Firstly, I would like to again clarify that it is possibly a coincidence that the metalloids start from boron, not carbon or any other non-metal in that period. My guess could be the start of electron occupation of the p subshells. Note that boron has one electron in its p subshell. Thus, its chemical properties start to change to become more non-metallic in nature. 

Diagonal relationships is a common occurrence in the Periodic Table. However, it is often used to describe the relationship between two elements which are diagonal to each other, not a whole line of them. Thus, I would like to clarify again that this is just my conjecture, not what is the true reason. But I believe this should be part of the correct explanation as it is logically sound. 



This site offers a more in-depth explanation as to why diagonal relationships occur:
https://chem.libretexts.org/LibreTexts/University_of_Denver/Chem_2132%3A_Chemistry_of_the_Elements/Unit_3%3A_Descriptive_Chemistry/09%3A_Building_a_Network_of_Ideas_to_Make_Sense_of_the_Periodic_Table/9.3%3A_The_Diagonal_Effect

In short, the diagonal relationships are based on two effects: the increasing effective nuclear charge across a period and the decreasing effective nuclear charge down a group. Effective nuclear charge increases across a period because the number of shielding electrons in the inner shells is constant while there are additional protons in the nucleus of the atoms of the elements across the period. Effective nuclear charge decreases down the group as there is an addition shell of shielding electrons from one element to the next in the group. (These are similar to the trends in metallic character pointed out in Pritt Balagopal's answer.)

These two competing effects cancel each other out and thus, chemists believe that the chemical properties of elements in a diagonal should be similar. The image shows some examples.