The war started a good 2, 3 years prior to actual hostilities commencing.

Both sides (we really can never say which side started it first, even today) did what they had been doing for a decade back then, and are still doing today - low level border outpost skirmishes. Basically, one side lobs a couple of mortar shells at some outposts on the other side of the border, the other side responds, and then some Jawan (soldier) fires an LMG / SMG, and there is a counter response, a few soldiers are killed on both sides, a lot more injured, gets escalated up the line, and a sort of a truce is settled till this cycle flares up again.

In 1965 though, the pattern shifted (to something that has been happening on the Indo-China border for decades now), instead of lobbing shells and exchanging gunfire, there were actual incursions into the border of the other country (again, if you ask the Indians, they will say the Pakistanis started it and vice versa), but sometime in Dec 64, Jan 65, Pakistani troops patrolled Indian held territory, and the firefights began to escalate.

The British stepped in and arbitrated the conflict and awarded a small parcel of land to the Pakistanis and both sides retreated to the pre skirmish lines.

Emboldened by this small victory and territorial gain, the Pakistani establishment felt that it could enlarge operations, and before the Indian Army could react properly, get external observers in, and settle for small territorial gains, while at the same time it could use covert troops in Kashmir to stoke up an insurgency (Pakistan first tried this in 1947, and has been doing it to this day). Thus began Op Gibraltar - the Pakistani attempt to sneak in large units of covert operators (we aren't talking small Seal Team 6 type operators, but more like the Resistance / Partisan type irregulars, in huge numbers) under cover of mortar and arty bombardment by the Pakistani military.

The Indian Army reacted, and quickly fought the infiltrators to a stand still, and then taking the offensive into PoK (to act as a buffer). Pakistan decided to then up the ante by sending in regular units to shore up the flagging covert op, however the Indian administration took the Pakistani establishment by surprise and broadened the war, and invaded the plains of Pakistan which is when the war officially begins.

One interesting, technical analysis of the war is David Higgins' book on it (I ran across this book when it itself was sourced on Wikipedia).