The "how" part is training on exemplars. First there needs to be some standard, and that was established by Daniel Jones with the cardinal vowel system. Essentially, Jones' students learned to produce the cardinal vowels, then taught their students. This provides reference points for a subset of the vowels in the IPA. The training tradition still exists in some institutions, and thanks to the miracle of the internet, one can find authoritative samples of the vowels (and consonants) performed by experts in the IPA, this chart from John Esling, and this chart with performances by Esling, House, Ladefoged and Wells (the Esling samples may be identical to his stand-alone collection). There are other such recordings, and classes that you can take.
These define the standard reference points for the IPA. If you hear Norwegian du you have a few choices as to which vowel it is closest to: credibly, [ʉ y Y] but not [u] or [æ]. If you listen to all of the reference samples, you will notice that the experts do not agree – this is a fundamental limit on the precision of the standard. Conventionally, "u" is assigned the value [ʉ] because (1) there is also [y] as in [dy], and (2) the vowel is auditorily closest to the standard [ʉ] for most expert performances. Each IPA symbol represents a range of values, and unfortunately ranges overlap a bit.
There is also a practice of micro-nudging vowel letters in the trapezoid, where the vowel [i] might be below the top line. This is the language-expert's judgment that the vowel is "a but lower", "a bit further back". Theoretically, this judgment can be aided by synthesizing vowels based on the formant values implied by this chart, and Keith Johnson did actually devise a "click on the chart" program which provided stimuli that aided people in making judgments as to where that vowel "is" (AFAIK it doesn't exist anymore for hardware reasons). One can generate comparative stimuli with Praat, which can give you a basis for saying "further back".
This is all about the phonetic values of vowels, and phonetic values are generally taken to be the primary determinant of vowel classification. However, phonological patterns also determine how people transcribe vowels. This is true to the point that people often disregard the phonetic facts in order to present a phonemic transcription, rather than a phonetic transcription. In Tigrinya, for example, the phoneme /ʌ/ is pronounced [ʌ ɛ æ a ɵ ɔ] depending on context, but this is (almost) never indicated in transcriptions, because it is rule-governed and people typically transcribe only contrastive details.
In other words, transcriptions and vowel charts are usually not claims about pronunciation, they are claims about the phonological system, but those claim are influenced by ideas of what the phonetics is. A couple of examples of phonetic vagarity are that the phoneme /e/ in Norwegian is usually a diphthong on the order of [ɛə], and the phonetics vs. phonology of Russian ы is a matter of extreme controversy (IMO [ɨ] is phonetically wrong but might be okay phonologically).
A chart like this might also represent mean formant values from a corpus in the language. Your particular chart looks more like a plot of computed formants, not a chart of auditory space.