The multiplicative inverse is the formal backbone of the arithmetic operation known as "division". I hope you can see that division of numbers has been and still is a quite significant thing.
"Reciprocal" means "multiplicative inverse", and it only makes sense to write the division symbols "1/2" when the reciprocal of 2 exists (in the ring).
A number in $\Bbb Z/n\Bbb Z$ has a multiplicative inverse iff $1$ appears in its row of the multiplication table.
One method is simply the extended Euclidean algorithm: \begin {align*} 31 &= 4 (7) + 3\\ 7 &= 2 (3) + 1. \end {align*} So $ 1 = 7 - 2 (3) = 7 - 2 (31 - 4 (7)) = 9 (7) - 2 (31)$. Viewing the equation $1 = 9 (7) -2 (31)$ modulo $31$ gives $ 1 \equiv 9 (7)\pmod {31}$, so the multiplicative inverse of $7$ modulo $31$ is $9$. This works in any situation where you want to find the multiplicative ...
So the multiplicative inverse of 1 is 1, the multiplicative inverse of 2 IS 4, the multiplicative inverse of 3 is 5, the multiplicative inverse of 4 is 2, the multiplicative inverse of 5 is 3, and the multiplicative of 6 is 6 (all "mod 7").
As for the question " What does it mean to be multiplicative inverses " Let $ (\Bbb F,+,\times)$ be a field (in our context, $\Bbb F$ can be the complex numbers and $+$ and $\times$ are the addition and multiplication we are used to)
2 I came across this sentence in J. Gallian's book - "Integer a has a multiplicative inverse modulo n iff a and n are co-prime/relatively prime" What is meant by "multiplicative inverse modulo n" of a number ??