It's not true so much now, but it was much truer 20 years ago. (And for the sake of the 20-year rule, I'll end the story in 1996.)

In both the Old World and the New, the "inner city" starts with industrialization. In places like New York, Boston, and London, a combination of rural depopulation/immigration (depending on your continent), and a huge need for unskilled labor in the manufacturing cities led to massive, largely unregulated expansion as the cities struggled to keep up with the demand for housing and infrastructure.

To illustrate: in 1811, the government of New York City laid out the city's grid plan as far as 155th St., roughly 10 miles north of City Hall. They believed that the grid would be sufficient to cover all future extension of the City. They were wrong. By 1879, the first rapid transit line had reached 155th St., and within another 20 years New York City would absorb four full counties (now the boroughs of Staten Island, Brooklyn, Bronx and Queens) into itself. Conditions in these tenement neighborhoods were horrifically crowded, polluted, and unsanitary by modern standards. For a personal anecdote, the house I grew up in was a 2-bedroom, 1-bathroom rowhouse of 1000 sq ft, and the previous owners raised a family of 9 there.

Clearly, something had to be done about this. In the States, it was thought that technology provided the answer. In the 1920s, urban planners believed that slum overcrowding could be alleviated by the wholesale destruction of those neighborhoods, the construction of new suburbs in outer-ring counties, and massive freeways to transport workers in and out of the cities. Conveniently, the neighborhoods targeted for destruction were largely poor and inhabited by ethnic minorities. For example, Black Bottom in Detroit, Boyle Heights in Los Angeles, and Little Manila in Stockton, CA were all shredded by the freeways as the U.S. went all-in on highway construction. This only made urban decay worse as businesses and remaining whites departed en masse. (Minorities were often excluded from moving to better neighborhoods by real estate brokers, assisted by the government-- a process called "redlining".)

The approach taken by European urban planners was totally different. Generally speaking, Europeans also attacked their social problems, but not with the kind of blunt instruments that the Americans used. This is partly because European cities were destroyed by war, but also because the dividing line in European politics was class rather than race, meaning that American-style demolition was never politically tenable. Instead, European governments built ring-roads around the urban cores, and left the urban cores to modernize largely organically. For comparison, here's Madrid, versus Detroit at the same scale. Notice how Madrid's M-30 ring road avoids the central city entirely, including most of the old residential neighborhoods. In contrast, Detroit's freeways deliver traffic directly into the downtown core, and the freeways run directly through the inner-ring residential neighborhoods, forming a complete loop. This was a disaster because it disrupted poor but functional neighborhoods, and uprooted them entirely. Anyone who could get out, did, leaving only the poorest and most desperate in the inner city.

For what it's worth, the major American city cores that avoided large-scale freeway-induced self-destruction have survived much, much better than their cousins. (Not coincidentally, the quintessential examples of this are San Francisco and Manhattan.) There are whole books written on this phenomenon, FYI.

So, in short: massive urban highway construction in the US led to urban decay; the lack thereof in Europe meant that their city cores survived better.

Jacobs, The Death and Life of Great American Cities; Caro, The Power Broker; Shoup, The High Cost of Free Parking; Gallagher, The End of the Suburbs; Kunstler, The Geography of Nowhere

It should be noted that there are some significant European cities, especially in the UK that have experienced urban decay in central areas and the flight of those with the means to the periphery, though most of them have tended to have seen large amounts of investment over the last 30 years and have significantly recovered today.

Nearly all the larger cities in the UK experienced issues with decay of the central city and surrounding inner city areas following WW2. This was for similar social reasons to decline in the US - people escaping smaller, high density housing and industrial pollution for new suburbs and surrounding countryside thanks to increase in popularity of the car.

Since the 80s most major cities in the UK have seen massive redevelopment of old commercial and industrial buildings and brownfield sites into residential usage, and a transition to more of a cafe/bar/nightlife focus in commerce.

London mostly escaped urban decay in central city (centres of older major suburbs weren't so lucky) with the exception of East end and docklands district (now a major business area with most of the cities skyscrapers), which used to be a very poor area with high crime and many derelict buildings until the mid 80s.

It has relatively little to do with urban geography at all, and it's not a synonym for "poor and crime-ridden" so much as it's a euphemism for "black." The central document for the stereotype itself, as distinct from the phrase, in sociological and policy discourse is of course the Moynihan Report of 1965, in which Senator Daniel Patrick Moynihan infamously reported that "the Negro family in the urban ghettos is crumbling." The specific "inner-city" euphemism dates back to the 1970s, but it really took off in the welfare-reform debates of the 1990s, when "centrist"/Third Way liberal sociologists like William Julius Wilson (e.g. When Work Disappears) gave the phrase the academic imprimatur.