The choice of a particular date as the beginning of "modern history" is ultimately one that is somewhat subjective and influenced by the particular national/regional/political context to which it is being applied.
However as a historian focussing on Europe, I will answer it from that perspective so please be aware this answer is but one of many.
1815 marks roughly the date of the Congress of Vienna, an international summit which was held between the great powers of Europe in reaction to the crises which arose following the French Revolution and the Koalitionskriege (French Revolutionary and Napoleonic Wars). Europe had previously seen almost a quarter century of near continuous warfare, and a longer history of conflict on the continent. The Congress hoped to create the conditions for peace in Europe, and to create the stability required to preserve the existing regimes. This was hoped to be carried out through reshuffling Europe into creating a 'balance of power' which would hopefully stop conflicts arising from any one state gathering too much power, or acting belligerently.
This system lasted more or less up to the First World War. You may have heard people talking about the war beginning as an effort by European nationstate trying to 'maintain the balance of power', and if so, this is broadly referring to the international system which arose after Vienna.
However, this is not the only choice for this date, personally I would date it to 1789 and the French Revolution. Largely this is because the events of 1815 were broadly a response to those caused by 1789, and further because while Vienna had a significant impact on international politics, 1789 had a greater influence on domestic affairs.
The 1789 revolution overthrew the Bourbon monarchy, paving the way for the Republican and Napoleonic systems. These systems saw the rise of mass politics, nationalism, conscription and bureaucracy (the Napoleonic Parlement is still used as an organisational unit in France today). Further the revolution had repercussions outside of France. For example, in Britain the Act of Union between Britain and Ireland was in large part a response to the fear of French invasion. Similarly the growth of the British state, with an expanding bureaucracy and system of taxation was a response to the subsequent wars against France. These changes, and the growth of a British national identity caused by the war helped pave the way for further changes in the British state.
Some good reads on this topic:
Robert Gildea, Barricades and Borders: Europe 1800-1914
Richard J. Evans, The Pursuit of Power 1815-1914