The below is just a quick summary and not too detailed, but I hope it answers your question.
As you say, Denmark did not participate in the First World War but gained territorially nonetheless. In fact, Denmark was the only country which remained neutral throughout the war to gain territorially from the Treaty of Versailles.
The territory that Denmark gained was Northern Schleswig. Denmark was granted this territory after a plebiscite in February 1920, in which 74% of the territory's population, being mostly Danish-speaking, voted to return to Denmark.
The reason this territory with a Danish-speaking population was controlled by Germany up to this point, was due to Prussia's invasion of the duchies of Schleswig and Holstein in 1864. Prior to 1864, the King of Denmark was also the Duke of the duchies of Schleswig and Holstein. These two duchies were primarily German-speaking (which is what led Bismarck to occupy them in his first steps to unifying Germany), however a Danish majority continued to exist in this northern part of Schleswig which now became a part of Prussia.
The Danish population in Northern Schleswig underwent oppression by Prussian/German authorities, being disallowed from being taught education in the Danish language, and being conscripted into the German army in the First World War. These Danes were promised, by the 1866 Treaty of Prague, a plebiscite to determine if they wished to return to Denmark, however this plebiscite was never held.
The Danish government obviously had a desire to regain this territory, but it was not something which the government was willing to compromise Danish neutrality over. After Denmark's catastrophic defeat to Prussia in 1864, Denmark viewed any military confrontation with Germany as futile. To a certain extent, the Danish government believed the best way to re-gain northern Schleswig was if it were rewarded the territory by Germany for remaining neutral during the war, as the Germans occasionally hinted at.
Since 1864, there had always been sympathy in Britain and France for Denmark and for this Danish minority in northern Schleswig. Many viewed Denmark as the first victim of "Prussian militarism", and believed that Denmark's defeat in 1864 had to be rectified.
Whilst there were some in Britain who argued that Denmark should join the Allies in order to regain Schleswig, military planners wrote off this idea due to, amongst other reasons, the weakness of the Danish military, as well as the clear lack of support for this idea from the Danish government, being more of a hinderance to the Allies than a benefit.
Regarding post-war territorial settlements, Britain as early as 1916 began considering granting Schleswig (not just the northern part) to Denmark after the war. This was not only due to the Allied belief of "the right to determination", but also of course to weaken Germany. Primarily France, but also Britain, favoured the Danish border being moved as close to the Kiel Canal as possible, so that in any future wars with Germany the Allies may more easily threaten this important canal.
Denmark was officially invited by the Allies in 1919 to send representatives to Versailles to represent their case for re-gaining Schleswig. These Danish representatives however, actually asked for less than Britain and France were suggesting. Whereas Britain and France were willing, to certain extents, to grant the entirety of Schleswig, Denmark requested that plebiscites be held in Schleswig and only those territories voting in favour of joining Denmark be granted to them. The USA, of course, backed the "right to determination" and did not greatly care about Allied concerns regarding the Kiel Canal, and therefore backed the Danish proposal.
Schleswig, for the purpose of the plebiscites, was divided into three zones: a northern, central, and southern zone. The southern zone clearly had a German majority, and therefore no plebiscite was even held here. As said before, the plebiscite in northern Schleswig returned a 74% majority in favour of joining Denmark. The plebiscite in the central zone returned an 80% majority for staying with Germany, and was therefore not granted to Denmark. Northern Schleswig was officially absorbed by Denmark in June 1920.
The border established in 1920 remains the Danish border today. Interestingly, Northern Schleswig was the only territory which Germany lost in the Treaty of Versailles which Hitler never attempted to reverse.
Source: Kaarsted, Tage, Great Britain and Denmark: 1914-1920 (Odense: Odense University Press, 1979)