The main reason is that you're looking at phonemic rather than phonetic transcriptions. IPA can be used for both; the way you usually distinguish is that you use slashes for phonemes, brackets for phones.1 So your linked example, /səˈlɪs.ɪt/, is phonemic.
A phoneme isn't a sound, it's an equivalence class of phones used by a particular language.
For example, consider the English phoneme /k/, as pronounced in "kit" /kɪt/ and "skit" /skɪt/. The first is an aspirated voiceless velar plosive, [kh]; the second is an unaspirated voiceless velar plosive, [k].
Why are these considered the same phoneme? Technically, it's because there's no English pair of words that's distinguished by one having [k] and the other having [kh]. But the intuition that the technical definition is trying to explain is pretty simple: Native English speakers don't even think of them as different sounds. It's hard to hear the difference unless you stop listening for speech and start listening for raw sound, and it's hard to pronounce [kɪt] instead of [khɪt] even after you recognize the distinction.2
But in many other languages, like Hindi, they are separate phonemes, because there are words that are distinguished by having /k/ vs. /[kh]/.
This is called "allophony", and it's just one of the reasons the same phoneme can have two different sounds.
Accents are another reason the same phoneme can sound different. The usual dramatic example is John Lennon pronouncing "bottle". All English accents allophonically vary /t/ intervocalically, but some Northern English accents vary it to [?], which to speakers of many other accents "doesn't sound like a t" or sometimes even "doesn't even sound like a consonant".
Less obvious reasons include things like pitch. Some languages (e.g., Cantonese) have syllabic tones, which are usually considered part of the phonemic representation. But in most languages where pitch doesn't mean tone, it means something else—something that varies across words, or phrases. For example, in English, questions are distinguished by rising tone in the last word, which affects the sounds exactly the same way Cantonese tones do, but isn't considered part of the phonemes.3
And of course even a phone isn't a sound.
This is most obvious from the fact that a person with a high-pitched version and a person with a low-pitched voice can both say /skɪt/, and they're both recognizably saying [skɪt], but they sound totally different.4
1. Alternatively, depending on context, slashes are used for the broadest phonetic transcription and brackets for a narrower phonetic transcription. The details are different, but it's easier to explain the categorical difference and then ask you to extend it in the obvious way to the degree difference, than to explain the continuous degree difference first.
2. For example, you may have to hold your hand in front of your mouth so you can feel the difference, until you can finally recognize what you're doing and control it.
3. Famously, stereotypical girls of the previous generation used the same pitch contour even on many non-questions, which drove lots of older people to grumble about the kids today. Now those kids are grown up and grumbling about why their kids use vocal fry in all kinds of places it doesn't belong. It never takes much to get one generation to grumble about the next one's language.
4. To understand how they can both be heard as saying [ɪ], you just need to learn about formants; it's not as complicated as you might expect.