Actually the interests are not much different from the answer of this question, but I will try to explain the perspective of the Turkish government.
Visa-free travel is not Turkish people's first priority for accession to the EU, but only the third according to the Economic Development Foundation's report. If President Erdogan and his party respected people's will, they would try to accomplish the first two wills.
- The level of prosperity and economic development
- Development of democracy and human rights
- Roaming, settlement and education in Europe
I can list the following reasons for the Turkish government's insist on visa exemption.
1. Turkey is a member of the Council of Europe*
According to the agreement in 1957, Turkish citizens already had the right of visa-free travel. The agreement was in force until 1980, but Turkish citizens were excluded after the 1980 Turkish coup.
In 2013, Erdogan (was Prime Minister at that time) blamed the military junta in Turkey for that. The translation of his word is as follows.
The visa application was demanded by the coup regime after September
12, 1980. In particular, visa restrictions were put in place to
prevent the intellectuals and artists of our country from escaping
from the pressure of the junta.
2. Turkey is an equal candidate for accession to the EU
The EU recognized Turkey as a candidate for full membership in 1999. Erdogan thinks that this should have led a visa exemption (from the speech in 2013 again).
The visa hurdle should have been removed much earlier. Turkey should
benefit from these opportunities as other candidates.
3. Turkey-EU relations are getting worse
The Turkish government wants to restore relations, which didn't go well in the recent years. Erdogan says that the visa matter is a good opportunity for that.
Turkey-EU relations suffered a loss of momentum. This is not caused by
Turkey as all our friends know well. Our relations will gain momentum
by the recent opening of negotiations on 22nd chapter and these