Since $\frac{\phi(n)}n=\prod_{p\mid n}\frac{p-1}p$, where the product is over all distinct prime factors, the lines are almost through the origin and mostly exhibit the distribution of distinct small prime factors in $n$, where the eye does not distinguish the differences between the factor $\frac{p-1}p$ for the largest prime factor, which is usually different between the values of $n$ on the same "line".
So the topmost line is for the prime numbers themselves, where $\phi(n)=n-1$, is a straight line almost through the origin; the "halfway" line is for numbers $n=2^kp^l$ (for some relatively large prime $p$); it is almost equal to the line through the origin with slope $\frac12$. In fact it is an almost straight line $\phi(n)=\frac n2-\frac n{2p}$ that holds for such numbers, with downards dents when the final factor $p$ is not so very large. Other "lines" can be explained similarly, but are less and less marked. And none of these lines are straight; if you look carefully there are points that tend to "droop".
Added: in fact one can explain these non-straight lines also as a union of several (in principle infinitely many) very close straight lines and lots of other points nearby. For instance the "line" for values $n=2^kp^l$ can be split into one for $n=2p$, where $\phi(n)=(p-1)=\frac n2-1$, one for $n=4p$, where $\phi(n)=2(p-1)=\frac n2-2$, ..., one for $n=2^kp$ where $\phi(n)=2^{k-1}(p-1)=\frac n2-2^{k-1}$ (which is still relatively close to $y=\frac n2$ when $p$ is very large), and other with higher powers of $p$, for instance $n=2p^2$ given $\phi(n)=p(p-1)=\frac n2-\sqrt{\frac n2}$ which is not a straight line but still relatively close to the line $y=\frac n2$. Indeed even other points would contribute visually to a clustering around (or more precisely just below) the line $y=\frac n2$: when $m=2pq$ is twice a product of two large primes $p, then $\phi(m)=(p-1)(q-1)$ is less than $2q=\frac mp$ away from that line. The same occurs by the way just below the line $y=x-1$ containing $(p,\phi(p))$ for any prime $p$: any number that only has few and very large prime factors will produce a point just below that line.
In the given graph I can only distinguish four lines clearly: the top one with slope $1$ for the prime numbers, the line with slope $\frac12$ for the numbers with only $2$ as small prime factor, the line with slope $\frac23$ for the numbers with only $3$ as small prime factor, and the line with slope $\frac13$ for the numbers with only $2$ and $3$ as small prime factors. Knowing the phenomenon one can discern lines with slopes $\frac45$ and $\frac67$, but already these don't stand out very clearly.