The bonds represent your stake in the country. Whoever has invested the most in a country (has the highest total cost of the bonds of that country) runs the country. Ownership can only be passed when another player exceeds the current owner's investment.
When the country lands on the Investor space, money is paid out from the country's treasury to the shareholders. There must be enough money to pay out the interest to all the non-owner players who hold bonds in that country. The owner then takes his interest out of the remaining (if any) money.
Once a country reaches 25 on the score track, each player's bonds pay out their interest x the country's multiplier on the track, adding to their final score.
Basically, buying bonds gives you a share in the country, having the most gives you ownership of the country, and in the endgame gives you points for good investments, if those countries did well.