Since the UK has first-past-the-post system for MP elections, when it comes to MP elections, it's important to accurately predict seat-level results, not merely the overall popularity of a party.
Unfortunately, running conventional large-sample polls for every seat is very cost-prohibitive, so this is not done. A typical survey poll needs around 1,000 respondents. Multiply that by 650 seats and you'd need poll over half million people for seat-level results in conventional way...
So MPR is way to work around that and still get seat-level predictions. As Wikipedia explains:
Multilevel regression and poststratification (MRP) is a statistical technique used for estimating preferences in sub-regions (e.g., states, individual constituencies) based on individual-level survey data gathered at other levels of aggregation (e.g., national surveys).
The "trick" is to poll fewer respondents (for example the last YouGov MPR poll only surveyed some 13,000 people, still an order of magnitude more than for a regular popularity poll, but a lot less than would be needed for a conventional poll in every seat) but to compensate for that by having a historically and/or census-informed model of how set-level results differ from the mean national results. As YouGov explains:
There are not enough observations in individual constituencies to treat the data as separate constituency polls, but we can
look for patterns in responses across constituencies that have similar characteristics, and then work out the implications
of those patterns for each constituency. [...]
Our modelling approach proceeds in two steps. First, we build a detailed description of which kinds of people live
where in the UK. This “population model” tells us how many people we think voted in different ways in recent elections, are
of each age, gender, level of educational qualifications, etc, and all the combinations of these, in each UK parliamentary
constituency. The population model is constructed from a number of sources, including past election results, a variety of
types of survey data and UK census data.
Second, we use the YouGov survey data described above to build a survey response model of current voting intention
(or of responses to any of the questions listed below). This is a model of how voting intention is associated with the individual
level characteristics that are in our description of the population plus characteristics of parliamentary constituencies. The
survey data might indicate that people with different educational qualifications are giving different patterns of responses,
or that different parts of the UK are giving different responses, or other similar patterns
To make up a simple but fictitious example that only uses census data (based on one of the papers linked in Wikipedia), assume you have a country with two political subdivisions, one which has more younger people than the other. Then an MPR poll would note the repose (to some question of interest, e.g. preferred party) as well the age of the respondent. Then the first step in an MPR is to build a regression model (typically logit) that predicts the outcome based on the age of the respondent (at national level). The second step is to use (in this case, only) census data to predict constituency-level outcomes using the same coefficient [for age in this case] obtained at national-level, but applying it to constituency-level data [from the census]. So, in this simple model, seat predictions could differ between two regions if one has more younger people than the other region, if age is substantial differentiator of party choice (which often enough it is.) Of course, such a model can be enriched with more relevant variables beyond age.