In USA (and I'll confine my answer to there since that seems to be where your question is pointing) election terms the first great year is actually 1936 (as an amateur psephologist it's still a tale pollsters like to reference for the lessons in it, and the smugness). Where you get the two strands of opinion polling intertwining (the media and the science) for the first time.

On the media side there was a magazine called Literary Digest that had been running large scale unscientific polling for several elections, essentially just asking all its readers who they would vote for (and getting millions of responses, which they felt would lead to the best results).

In 1936 they confidently predicted that Alf Landon (Republican Governor of Kansas) would win the US Presidential Election.

At the same time you saw the first polling companies being formed by applying market research techniques into politics.

These market research companies (Nielsen would be a famous name here) had been formed in the early 1920s to better direct the relatively new era of mass advertising, (and also mass production) and also judge the size of radio audiences (because you suddenly had a medium without an easy way of finding out how many people were receiving, unlike say newspaper sales numbers).

So you had people like Elmo Roper forming these market research companies in 1933, while (the rather more famous) George Gallup was a director of research at an advertising agency (Gallup had an academic background in psychology and journalism, so was very interested in studying people's attitudes etc). Gallup got his political start doing some polling for his mother-in-law election victory (Ola Babcock Miller) in 1932.

In 1936 Roper was contracted by Fortune magazine to predict the election, Gallup had in 1935 formed his own polling company (American Institute of Public Opinion) and using "just" (it would still be a huge number compared to modern polls) 50,000 responses they both called the election for the Democratic incumbent, Franklin Delano Roosevelt.

Roosevelt (spoilers) won in a massive landslide of 523 votes to 8. And the pollsters had their major day in the sun, which was important since (and this is still mostly true today) the election polling mainly functioned as advertising for them to show off to their clients at how accurate they were (it is after all a rare occasion where you get a pretty definitive result to compare against). The victory helped solidify a very new industry and gave it a foundation to crow about.

Both organisations would later be called on by FDR to gauge public support for Lend-Lease during WWII.

There was a sizable wobble in 1948. By which time you had much greater polling going on, through the primaries as well as the election and where you first start to see how candidates were viewed being shaped by their poll performances, there was a movement to consider dropping Truman fuelled in part at least by his poor polling scores.

In the run up to the election Dewey was leading heavily, and most polling in fact stopped weeks before election day (with Truman gaining from a long way back). So the pollsters (and everyone) called it for Dewey, and (spoilers) in fact Truman won.

This led to a lot of ridicule of the industry, but they were entrenched enough to defend themselves through it (and the analysis they produced suggested that the error was stopping too early and Truman made great strides among people who didn't make up their minds until the final couple of weeks, which went completely against conventional wisdom at the time).

So polling was established as a major part of the election cycle by the early post-war period.

In terms of recent increase you talked about, this is certainly true and for a very good reason.

The thing about polling, is that it is generally really expensive. You have to get thousands of people to answer questions for you for every poll (and they're outdated in an election within a week).

Telephone polling is of course easier than face-to-face polling but you still have to employ, train, etc a lot of people to make all those phone calls. The big shift in recent years is internet polling, suddenly you can automate so much of the expensive parts into lines of code and make everything so much cheaper (also the computer power to calculate things for you helps as well).

The flood of polling in the last 10 years or so has been driven by a massive increase in internet based polling (how well this stacks up against telephone polling quality wise is still up for debate) enabling far more polls not just in frequency but variation. So we have the daily trackers nationally and state-wide, but also all kinds of smaller more specific polls for particular issues, groups etc.

So we have more (and probably better) polling than anytime in history, and also a media that between websites and 24/7 news media cycle desperate for something to feed on has created more niches for polling experts and analysts to slide into and gain more prominence. Equally with the increasing levels of numbers creeping into sports the public is probably more receptive to numbers in their coverage.

So in short, came out of market research in the 1930s (and political polling has generally been advertising for the profit-making commercial side of the business), and has had a recent boom driven by technology reducing costs.