Your questions could be answered from two different perspectives or by two different interpretation: A historical perspective related to time of the Prophet (peace be upon him) and the implication for us or every generation afterwards.
Who are the people he is complaining about?
Historically apparently all (sunni) mufassireen are in consensus, that the people our Messenger (peace be upon him) was complaining about where his people: the tribe of Quraish!
But this could also apply to any Muslim (after the time of the Messenger of Allah (peace be upon him) and now) who abandoned the Quran.
What is the meaning of abandoning the Qur'an?
Some of the meanings from the historical perspective were quoted by servant, but most could be found in the Quran itself:
They used not to prevent one another from wrongdoing that they did.... 5:79
They didn't advise each other or prevent each other from doing wrong.
And they prevent [others] from him and are [themselves] remote from him ... 6:26
They tried to kept themselves and others away from listening to the Quran
Do not listen to this Qur'an and speak noisily during [the recitation of] it ...41:26
They didn't listen to the Quran and tried to create disturbance, by speaking loudly, so that the believers also couldn't listen!
Imam al-Qurtoby quoted a hadith from a book of a-Tha'labi on the authority of Anas (May Allah be pleased with him) to explain abandoned:
من تعلم القرآن وعلق مصحفه لم يتعاهده ولم ينظر فيه جاء يوم القيامة متعلقا به يقول يا رب العالمين إن عبدك هذا اتخذني مهجورا فاقض بيني وبينه
From this we could conclude some meanings for hajr/abondoning for any later generation (some of the points are taken from my later reference from a quote of ibn al-Qayyim):
- They don't apply the rules of the Quran.
- They don't listen to it (which may also mean take it as a basis for rules).
- They don't believe in it (or reject parts of it).
- They don't base the ahkam of sahri'a on it.
- They read it but don't try to understand and reflect on the meanings of what they read.
- They don't read the Quran.
- They don't teach and memorize it.
- They left the Quran and toke other sources for religious ruling (from tafsir ibn Kathir)
- They don't seek cure for their disease and help in it.
- They just keep it as a decor unused in their houses.
One could summarize they don't make it a part of their lives! And this is in total contradiction to our Belief.
As we are asked as Muslims to follow the example of our beloved Prophet (Peace be upon him) about whom his wife 'Aisha (May Allah be pleased with her) said (see here):
His character was the Qur'an ...
What are the consequences?
Again I'd like only to add consequences that servant has not quoted.
We know from the Quran what happend to the people of book when they abandoned there books (7:169).
Note that the word خَلْفٌ in Arabic refers to succession or offspring who are from both kinds bad/sinful and good, while خَلَف refers to good offspring etc.
Imam al-Qurtoby (died 671 a.H.) in his tafsir of this verse said that this description already applies to us (Muslims).
On the other hand we have the opposite consequence if we don't abandon the Quran: According to this hadith from sahih Muslim (see also in sunan ibn Majah) our Messenger has said:
By this Book, Allah would exalt some peoples and degrade others.
which some of his Sahaba (May Allah be pleased with them) have seen later as described in the hadith above.
So the major consequence of abandoning the Quran is: losing the guidance and a weakness of our faith and believe and we may loose the right path and get lost in this dunya as other nations whom have been given guidance by Alalh did.
An Arabic article on the verse (25:30)
See also tafsir ibn 'Ajibah البحر المديد في تفسير القرآن المجيد