更 means "replace". 岁 means "year". 交 means “meet, at". 子 means 子时, the first of the 12 two-hour periods. It starts at 23:00 and ends at 01:00. In the traditional time system, one day is devided into 12 equal parts, each of which lasts two hours. They are named after the 12 Earthly branches(地支). The first starts at 23:00 of 24-hour system used today. It is the beginng of one new day. So 更岁交子 means that the new year replaces the old one at the beginning of the first hour (when the time meets the first hour). Besides, 交 and 子 together is just right a word 交子. It is a kind of currency used in the Song dynasty and therefore it means money. What's more, 交子 pronounces the same with 饺子(jiaozi, dumpling). In China, people eat jiaozi on the last day of the Lunar Year. Eating jiaozi is a family activity which has the meaning of good luck, happiness and fortune. 更岁交子 is used to express that the new year comes and wish everyone good luck, happiness and fortune.
Since there many characters or words that sound the same or similar in Chinese, the art of homophonic is used quite frequently.