I've copied below the wikipedia algorithm for computing the LPS array (which is usually called the failure function because in the main KMP algorithm you refer to this table whenver there's a failed match). They denote $T$ where you use $LPS$
an array of characters, W (the word to be analyzed)
an array of integers, T (the table to be filled)
nothing (but during operation, it populates the table)
an integer, pos ← 1 (the current position we are computing in T)
an integer, cnd ← 0 (the zero-based index in W of the next character of the current candidate substring)
let T ← -1
while pos < length(W) do
if W[pos] = W[cnd] then
let T[pos] ← T[cnd]
let T[pos] ← cnd
let cnd ← T[cnd] (to increase performance)
while cnd >= 0 and W[pos] <> W[cnd] do
let cnd ← T[cnd]
let pos ← pos + 1, cnd ← cnd + 1
let T[pos] ← cnd (only need when all word occurrences searched)
The running time will be dictated by how many times "cnd" gets assigned to. The while loop inside the for loop ($pos$) might make you think the algorithm is $O(m^2)$ but as you learned it is $O(m)$.
Inspect how $cnd$ may change during one step of the for loop; it can either increase by +1, or it can decrease inside the while loop. And when it decreases, it decreases by at least 1, since $T$ satisfies $T[x]<x$ for all $x$ ("proper" in the definition of LPS). Since $cnd$ can increase only by +1 and there are $m$ steps in the for loop, this means the overall increase is bounded by $+m$. You might visualize each change to $cnd$ as a stair of integer height, with upstairs and downstairs representing increases and decreases to $cnd$; the upstairs are all height 1, and the downstairs are each height at least 1. Because the overall height never falls below 0 (that is the termination condition for the while loop), you can bound the total number of stairs by $2m$; there are at most $m$ "up" stairs and there can't be more "down" stairs than up stairs since their height is $\geq 1$.
To put another way, this is basically a twist on the saying "what goes up most come down"; here we have "what comes down must have gone up before"