Why is the Thai language classified as Sino-Tibetan/Sino-Burmese when its script looks like Sanskrit to me?
The script has nothing to do with the origin of the language. In fact, every script can be used to write any language. Usually a language adopts the script that is associated with the religion and/or dominating cultural influence. For example, Malay, which belongs to the Austronesian family of languages, used the Arabic script (with some variations) when the Arabic influence on the Malaysians was at its peak, but that doesn't mean they are a Semitic people. Now they use the Latin alphabet, because the Western influence on Malaysia is far more intense than previously, but that doesn't mean they are related to Romans or to Indo-Europeans directly.
As far as Thai is concerned, firstly, it does not belong to the "Sino-Tibetan/Sino-Burmese" languages, Thai is a member of the Tai–Kadai language family and it is not related to Chinese, although Chinese has influenced it a lot. When the Chinese influence on Thai was at its peak, Thai was written in Chinese characters. But later, in about the 13th century, the Thai people came under the cultural influence of Khmers who were Buddhists. The Thais converted to Khmer-style Buddhism and borrowed their script, having transformed it a bit. All the southern Buddhists like Khmers or Thais use scripts that come from India; those scripts are all descendants of the ancient Indian script Brahmi.
Sanskrit, being one of the languages the sacred Buddhist scriptures were written in, also greatly influenced Thai, which has lots of loan words from Sanskrit. Still, neither borrowed words nor borrowed scripts make languages related. To see the relationships between languages is much harder than noticing similarities in scripts.
The modern Thai script is descended from Pallava, itself descended from Brahmi. It is similar and draws from Khmer, and is closest in similarity to Lao. Note that it is largely unrelated to the Myanmar/Burmese script. Sanskrit is itself is a language written in a variety of scripts, including Thai). On the other hand, Pali (the liturgical language of Theraveda Buddhism) can be written in Thai, with the addition of three specifically Pali characters found in Thai fonts (Nikkhahit, Pinthu, Yamakkan). On the other hand, Thai has many loanwords from Pali and Sanskrit.
I see people have talked more on borrowing scripts to write words but less on the origin of words. When I look at Thai words(I am not a linguistic expert) I find a lot of words directly from Sanskrit. It looks as if Sanskrit had far great influence in Thailand area at some or most parts of last 1000 years or so which is also evident from Hindu(Sanatan Dharma) cultural influence in Thailand.
It' s not a sino language (I'm Thai and Chinese mixed, I know most of the Chinese dialects that exist and I know Thai and Lao) but Thai has MOST of its words from China, while the 2nd most words are from India (with the exception that the Thai words versions from India have tonals). The Thai script has nothing to do with the language btw. Thai and Cantonese for example share a similar pronounciation, it sounds similar and also has words who mean the exact same thing. There's a dialect spoken in south china that is very similar to Thai, even more then Cantonese.