I am working on my PhD in history of fashion right now. I focus on surf clothing, but from what I have found it is pretty similar to main stream clothing. People have paid for name brand clothing or designers since French Couture houses starting with the House of Worth (citation- Charolette Seeling, Fashion : 150 Years Couturiers, Designers, Labels).

But, post-WWII, is when clothing brands really started to put a logo, or trademark, on their ready-to-wear clothing. For example, the Hang Ten brand (one of the first board shorts companies) came up with the 2 golden feet on their board shorts and tee-shirts as a way for surfers to identify their clothing easily (especially in surf competitions). This happened in 1960. (Citation- Matt Warshaw, The History of Surfing).

Around this time, British brands were also putting a trademark or brand on their clothing. Think of the Burberry plaid pattern becoming extremely popular in the late 1960s. (citation-Harriet Worsley,100 Ideas that Changed Fashion.)

So shorter answer, it began earlier with high fashion and couturiers in Paris, but the actual logos became a staple in the 1960s.

edit: making the citations easier to spot edit 2: misspelled Worth

It's worth noting that at least in the US the reason for putting logos all over clothes is because the design of garments isn't patent or copyright-able so the only way to distinguish your product is by placing a trademarked logo on it. Here's some books that can tell you more if you're interested:

Between the Seams, A Fertile Commons by Christine Cox and Jennifer Jenkins - Duke Law Professor Jennifer Jenkins and intellectual property lawyer Christine Cox explore the relationship between fashion and various US intellectual property regimes, examining why fashion design generally is not protectable under copyright, design patent, trademark or trade dress.

Music & Fashion: The Balancing Act Between Creativity and Control by Aram Sinnreich and Marissa Gluck - Two scholars examine the subtle yet effective way the fashion community regulates copying, in contrast to the dense copyright protection and brute legal force that characterize the music industry.

Fashion Thinking: Towards an Actionable Methodology by Natalie Nixon and Johanna Blakley - An explanation of what "fashion thinking" is and how it can inform innovation for a broad range of consumer products and services.

As another poster mentioned, Lacoste started putting crocodiles on their polo shirts in 1933. The story starts with René Lacoste who won seven singles tennis grand slam titles in the 1920s. He was nicknamed "the Crocodile" because of his playing style and his reputation as a hard-working player who made few mistakes, studied his opponents, and dominated the tennis world of the 1920s.

In 1926, Lacoste asked a friend to design a crocodile logo which was stitched onto one of his blazers (image), but it was still only a personal brand until later in 1933 ^[1]. René Lacoste co-founded La Chemise Lacoste in 1933, the logo was placed on their polo shirts (which Lacoste was known for wearing on the tennis court), and the company grew into a huge brand over the coming decades.

But, the Red Diving Girl beat the Crocodile to the punch by a decade.

Jantzen is a swimwear brand started in 1916 (based in Portland, Oregon) ^[2] and they definitely provide an earlier example in response to the question. The Red Diving Girl first appeared on a 1920 catalog cover and turned into an advertising icon starting in 1922 when the company printed 10,000 stickers, originally for store displays, but quickly landing on car windows. By 1927, over 5 million car decals had been printed and distributed. The Oregon Historical Project has a 2002 article going into more detail on the history of Jantzen and the logo.

When did the Red Diving Girl first land on a outwardly visible piece of clothing? It first appeared on Jantzen swimming suits as early as 1923. The Kobe Fashion Museum has a Jantzen swimsuit from 1923 with a visible Red Diving Girl logo ^[3]^[4].

As an icon, the Red Diving Girl is today not as well known now as the Crocodile, but Lacoste was not the first company to add a brand logo to their apparel. Of course, it's difficult to disprove that there wasn't a company earlier than Jantzen, but Jantzen was and continues to be a very well-known brand. The history of the Lacoste brand with a brief mention of the Red Diving Girl is discussed more in the Brand Bible By Debbie Millman.

edit: cleaning up some citations

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Beyond the logo of the brand of the clothing itself, when did clothing with other logos or advertising become popular? Not sure if that's what the OP was asking. Say, a sweatshirt with "Coke" on it, not the logo of the clothing designer or maker.

As a follow up, when did people first start pointing out that branded clothing was advertising? Any early backlashes?

For style and identity. Brands striving hard to maintain branding and day by day investing more by adopting new ways to brand marketing. Having a corporate logo own clothing is just another way to trigger branding.